J. Evol. Equ. (2022)22:58 © 2022 The Author(s) Journal of Evolution https://doi.org/10.1007/s00028-022-00816-4 Equations Semilinear formulation of a hyperbolic system of partial differential equations Carles Barril and Àngel Calsina Abstract. In this paper, we solve the Cauchy problem for a hyperbolic system of first-order PDEs defined on a certain Banach space X . The system has a special semilinear structure because, on the one hand, the evolution law can be expressed as the sum of a linear unbounded operator and a nonlinear Lipschitz function but, on the other hand, the nonlinear perturbation takes values not in X but on a larger space Y which is related to X . In order to deal with this situation we use the theory of dual semigroups. Stability results around steady states are also given when the nonlinear perturbation is Fréchet differentiable. These results are based on two propositions: one relating the local dynamics of the nonlinear semiflow with the linearised semigroup around the equilibrium, and a second relating the dynamical properties of the linearised semigroup with the spectral values of its generator. The later is proven by showing that the Spectral Mapping Theorem always applies to the semigroups one obtains when the semiflow is linearised. Some epidemiological applications involving gut bacteria are commented 1. Introduction The evolution law of many dynamical systems can be represented by a differential equation of the form x  (t) = f (x(t)) in which the dependent variable t refers to time. Roughly speaking, it is said that the system is well posed if every initial condition x0 moves unambiguously and continuously into the future in accordance with the evolution law and if the trajectories described this way are continuous with respect to x0 . The well-posedness problem is usually referred as the Cauchy problem. As it is well known, if the phase space X of the system is a Banach space, then the system is well-posed if the function f : X → X is Lipschitz. In addition, if f is differentiable, then the local dynamics around steady states can be studied in terms of the linearised system around them. Such results are really useful if X is finite dimensional, since in this case all linear functions from X into X are Lipschitz, which implies that all linear systems are well posed. However, if X has infinite dimension, there exist linear functions (or operators) that are not continuous from X into X and fail to be defined on the whole X , such as those appearing in partial differential equations. The main result characterizing the linear operators for which the initial Mathematics Subject Classification: 35L45, 47D06, 92D25 Keywords: Semilinear hyperbolic systems, Sun dual formulation, Linearization principle, Spectral map- ping theorem, Spatially structured populations. 0123456789().: V,-vol 58 Page 2 of 33 C. Barril and À. Calsina J. Evol. Equ. value problem is well posed is the Hille–Yosida Theorem. Unfortunately, a general nonlinear theory is still incomplete. Some work has been devoted to solve the Cauchy problem for different classes of systems (see, for instance, the classical monographs of Pazy [22] and Henry [13] for semilinear semigroups and Miyadera [21] for dissipative systems). Similarly, different strategies exist to determine the asymptotic dynamics around stationary points of these systems (e.g. [5,15,20,26]). This range of techniques reflects the richness associated to differential equations in infinite dimension, as well as the fact that some generality has to be lost in order to obtain more accurate results. This article is a contribution to the program summarised above. We focus on a class of dynamical systems (see (1.1) below) including the model treated in [2] to study the propagation of pathogenic bacteria inhabiting the guts of animals. Whereas in [2] the model is analysed in a rather formal way, the theoretical framework sustaining such analysis is developed here. This is done using some results about dual semigroups and their relations with evolution equations [6,7]. 1.1. A class of first-order PDE systems Let u and v be functions with domain [0, 1] × [0, ∞) taking values in Rn and Rm respectively. Let r be a Rk valued function with domain [0, ∞). Let g : [0, 1] × Rn × Rm → Rn f : [0, 1] × Rn × Rm → Rm h : Rm × Rk → Rk be differentiable functions, and let c : [0, 1] → Rm be a bounded function such that ci (x) ≥ 1 for all x ∈ [0, 1] and i ∈ {1, , . . . , m}, where we use an index i to denote the ith component of a vector valued function (such as u, v, r , g, f , h and c). Let  be a m × k real matrix and let diag(c(x)) denote the m × m diagonal matrix whose entries are given by vector c(x). Then consider the following system of first-order partial differential equations with initial condition: ⎧ ⎪ ∂t u(x, t) = g(x, u(x, t), v(x, t)), ⎪ ⎨ ∂t v(x, t) = −diag(c(x))∂x v(x, t) + f (x, u(x, t), v(x, t))  (t) = h(v(1, t), r (t)) (1.1) ⎪ ⎪ r ⎩ u(x, 0) = u 0 (x), v(x, 0) = v0 (x) and r (0) = r0 , with boundary condition v(0, t) = r (t). (1.2) The problem above is said to be well posed if, at least for small times t, it deter- mines unambiguously the trajectory of the system and such trajectories present some continuity with respect to initial conditions. In order to prove that this is really the case the problem can be interpreted as a semilinear evolution problem defined on a certain Banach space X , J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 3 of 33 58 ⎧ du ⎪ dt = G(u, v) ⎪ ⎨ dv dt = −diag(c)∂x v + F(u, v) , (1.3) ⎪ dt = H(v, r ) ⎪ dr ⎩ (v(0), u(0), r (0)) = (v0 , u 0 , r0 ) ∈ X and then try to proceed as in the case of ordinary differential equations. The first dif- ficulty one encounters relates to the fact that the operator defining the right hand side of (1.3) is not smooth enough (i.e. Lipschitz) to apply the Picard iterative scheme, being the differential operator ∂x the reason of that. To overcome this problem Am- mon Pazy’s book [22] develops a theory to treat the system composed only by the “difficult” (but linear) principal part (i.e. the differential operator or whatever that creates the loss in regularity) and after consider the whole system as a smooth per- turbation of the system generated by just the principal part. The tool to link the perturbed system with the unperturbed one is the variation of constants formula, which makes possible to prove not only the well posedness of the problem but also the principle of linearised stability. Although this is a typical procedure to deal with semilinear partial differential equations, in our case some difficulties arise due to the nonlinearities G and F involved, which take the form of Nemytskij opera- tors defined by g and f , namely G(u(·, t), v(·, t))(x) = g(x, u(x, t), v(x, t)) and F(u(·, t), v(·, t))(x) = f (x, u(x, t), v(x, t)). In fact, these difficulties were also high- lighted in [18] for a hyperbolic system similar, but not reducible, to (1.3). First of all, we have to decide which Banach space X we should use in order to study the system above. From a conceptual point of view, if u and v are densities on the interval [0, 1], it would be natural to consider spaces based on L 1 for these variables. However, the Nemytskij operators G and F defined on a space of integrable R-valued functions on [0, 1] are very often not well defined and more importantly, they are Fréchet differentiable only if g and f are affine functions [17], which prevents us from using well defined but truncated versions of G and F. This makes impossible to linearise the system around steady states in order to study their stability properties. Although this lack of differentiability does not invalidate the principle of linearised stability per se, it makes necessary to use ad hoc techniques to analyse the behaviour of the system around stationary points (see [10] for an example of that). In order to avoid this lack of smoothness related to the Nemytskij operators on L 1 , we may use spaces based on the sup norm. It is easy to prove that these operators inherit the smooth properties of their associated functions in spaces with the sup norm. In particular, if g is differentiable, then the Fréchet derivative of G at a point (ū, v̄) is the operator   u DG(ū, v̄) = D2 g(·, ū(·), v̄(·))u(·) + D3 g(·, ū(·), v̄(·))v(·), (1.4) v and analogously for f . Specifically, the sup norm space X we are going to work with is X = X1 × X2 (1.5) 58 Page 4 of 33 C. Barril and À. Calsina J. Evol. Equ. with X 1 = L n∞ := (L ∞ (0, 1))n and  X 2 = (C m × Rk )b := (v, r ) ∈ C([0, 1], R)m × Rk | v(0) = r . Realise that the definition of (C m × Rk )b comprises the boundary condition of the system. The reason why we choose X to have this form rather than a simpler one such as L n∞ × C([0, 1], R)m × Rk or L n∞ × L m ∞ × R is that the trajectories would not be k continuous on these spaces. In order to follow the standard semilinear formulation of system (1.3) as it is de- scribed in [22], we consider the linear principal part and the nonlinear Lipschitz per- turbation separately. Thus, the operator A defined as ⎛ ⎞ ⎛ ⎞ u 0 A ⎝ v ⎠ = ⎝ −diag(c(·))v  (·) ⎠ (1.6) r 0 with domain D(A) = {(u, v, r ) ∈ X | (0, diag(c(·))v  (·), 0) ∈ X }, is the infinitesimal generator of a strongly continuous semigroup T . The prime symbol  in the previous expressions refers to the generalised notion of derivative. Being the principal part of the system specified, the perturbation is the operator P that sends elements (u, v, r ) ∈ X into P(u, v, r ) = (G(u, v), F(u, v), H(v, r )). At this point, however, we encounter an obstacle that prevents us from applying the standard semilinear formulation in a straightforward manner. It turns out that the range of the perturbation P is not within X but contained in the bigger space Y = L n∞ × L m ∞ ×R . k Since P is Lipschitz from X into Y , the problem is now not a lack of regularity but a lack of definition of the semigroup T on the space Y , in the sense that the variation of constants equation ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ u(·, t) u 0 (·)  t u(·, s) ⎝ v(·, t) ⎠ = T (t) ⎝ v0 (·) ⎠ + T (t − s)P ⎝ v(·, s) ⎠ ds r (t) r0 0 r (s) is ambiguous. To overcome this difficulty the semigroup T should be extended into Y , and the natural way to do that is to define the operator A in Y instead of X . However, this procedure is not as easy as it could seem. In X some hypotheses are satisfied by A that guarantee the existence of a strongly continuous semigroup whereas in Y such hypotheses could not hold. In order to deal with semigroups that fail to be strongly continuous we apply the sun-dual framework developed in [6,7]. This theory extends the standard semilinear formulation by allowing perturbations that take values on certain Banach space related to the phase space in which the system is defined. It is worth pointing out that this theory has commonly been used to treat systems of delay differential equations [8,11], whereas here it is applied to a system of PDEs (similarly as in [18]). J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 5 of 33 58 This article is organised as follows. In Sect. 2 the theory of sun-dual semigroups is used to prove that the initial value problem (1.1) with boundary condition (1.2) is well posed (Theorem 1). In Sect. 3 a criterion to infer the (local) stability behaviour of a stationary solution of (1.1) is given (Theorem 4). This is done by showing that the linearised semigroup around any stationary solution of (1.1) is eventually norm continuous (Theorems 2 and 3), which implies that the behaviour of the linearised semigroup can be summarised in terms of the dominant eigenvalues and eigenvectors of its generator. In Section 4 we show that problem (1.1) and (1.2) can be used to model the growth of microorganisms along the intestines of animals, a topic with implications ranging from human health [23] to biodiversity [24]. The model is an extended version of the one model formulated in [1] and treated in [14,25]. The extension (first presented in [4] and later developed in [2]) includes environmental microbes, which makes possible to study reinfection events and transmission between hosts. In the same section we illustrate how the results of Sects. 2 and 3 are applied to this particular model. In Appendix A we recall the most important results (for our purposes) of the work done by Clement et al. [6,7], which can be used to study the existence and uniqueness of solutions to our problem as well as the stability properties of the steady states. Several definitions and results given in the appendix are used widely in Sects. 2 and 3. 2. Semilinear formulation of the problem To deal with problem (1.3) we first give a formula for the semigroup T generated by A (see (1.6)) and then we show that T is indeed closed by ∗-integration on Y (see Definition A2). This is done by checking that T has the structure of Example A1. Once this property of T has been verified, the well posedness of the problem will be derived as a consequence of Theorem A2. Finally, we show that the linearised semigroup S around any stationary point of (1.3) is eventually norm continuous, so that Theorem A7 gives a method to determine the stability of equilibria based only on the spectrum of the infinitesimal generator A S of S. 2.1. Existence and uniqueness of solutions The semigroup T on X generated by A can be obtained through the method of characteristics, and acts in the following way: ⎛ ⎞ ⎛ ⎞ u u(·) T (t) ⎝ v ⎠ = ⎝ ṽ(·; t, v, r ) ⎠ (2.1) r r where the ith component of ṽ(·; t, v, r ) is (ṽ(x; t, v, r ))i = ṽi (x; t, vi , r ) (2.2) = i r 1− (ϕi (−t, x)) + vi (ϕi (−t, x))1+ (ϕi (−t, x)), 58 Page 6 of 33 C. Barril and À. Calsina J. Evol. Equ. being i the ith row of  and ϕi (t, x) the unique function satisfying ∂t ϕi (t, x) = ci (ϕi (t, x)) and ϕi (0, x) = x (although function ci (x) is not necessarily Lipschitz, the uniqueness property can be deduced because ci (x) ≥ 1 for all x). Functions 1− and 1+ stand for the indicator functions on (−∞, 0) and [0, ∞) respectively. To give a meaning to ϕi (−t, x) for any −t < 0 is enough to assume that the function ci is prolonged by any positive constant (for instance 1) for negative values of its argument. Notice that ϕi (·, x) describes, as a function of t, the characteristic curve passing through the point x at time 0, so that ϕi (−t, x) should be interpreted as the position that a point moving with velocity c had t units of times into the past. Therefore, a point that at time t is at position x, was at position ϕi (−t, x) at time 0. In particular, if ϕi (−t, x) < 0 then we deduce that the point was outside the interval [0, 1] at time 0, hence the value of vi at (t, x) is not given by the initial condition of v but by the boundary condition. Remark 1. In order to keep track of the following results we recommend to consider the scalar case in which n = 1, m = 1 and k = 1. In this situation the subscript i can be dropped (since function v has only one component) and matrix  becomes a number. It may also help to consider function c to be constantly 1. Notice that in this case the function ϕ introduced above becomes ϕ(t, x) = x − t. Clearly, T is a diagonal semigroup on X = X 1 × X 2 = L n∞ × (C m × Rk )b (the detailed definition of X is given in (1.5)). Let T1 and T2 be the associated semigroups of T on X 1 and X 2 respectively. Since Y = L n∞ × L m ∞ × R , in order to show that T is k closed by ∗-integration on Y it is enough to verify that i) L m ∞ × R can be identified k T2 ∗ with a subspace of X 2 and that ii) X 2 is sun-reflexive with respect to T2 (as showed in Example A1). Notice that here Y has not exactly the same meaning as in Appendix A. There Y was a subspace of X ∗ whereas here Y is a representation of a subspace of X ∗ . We proceed in this way because then we can consider X as a subspace of Y and avoid the use of the inclusion j : X → X ∗ in the formulation of the results. Next we show that the two conditions mentioned above (i and ii) hold and, in addition, we specify how T2∗ is defined on L m ∞×R . k Proposition 1. Let X 2 = (C m × Rk )b and T2 be as above. Then k ∼ T2 ∗ T ∞ × R = X2 Lm and X2 ∼ = X2 2 . Moreover, T2∗ is the natural extension of T2 into L m ∞ × R , i.e. k     v ṽ(·; t, v, r ) T2∗ (t) = r r with ṽ exactly given as in (2.2). As a consequence, the semigroup T defined on X fulfills the hypothesis of Exam- ple A1 and hence is closed by ∗-integration on Y . J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 7 of 33 58 Proof. The methodology of the proof is based on the one used in [11, Section II.5], and hence we only expose the main ideas (a more detailed version of the proof can be found in [3, Appendix B]). First the Riesz representation theorem is used to represent X 2∗ as the space Mmb × R , where Mb is the set of real Borel measures μ satisfying k μ({0}) = 0. Instead of μ({0}) = 0 other conditions could be imposed in order to identify Borel Measures (seen as functionals on C([0, 1], R)m ×Rk ) that are equivalent on X 2 due to the boundary condition. The pairing between X 2∗ and X 2 can be written as      m  1 μ v , = vi (s)dμi (s) + q, r . q r 0 i=1 In order to work with functions instead of with measures, an isometric isomorphism between real Borel measures and normalised functions of bounded variation is used, so that X 2∗ is represented as NBVm b × R with η ∈ NBVb if η is a function of bounded k variation, right continuous in [0, 1] and satisfies η(0) = 0 (the norm in NBVb is the total variation norm). T The next step consists in showing X 2 2 ∼ = L 1 (0, 1)m × Rk where the norm in L (0, 1) is weighted by function c, in the sense that 1 m m     1  ν (s)  ν =  i   c (s)  ds. (2.3) i=1 0 i T This is done taking into account the result D(A∗T2 ) = X 2 2 (which is a particular case of the equality D(A∗T ) = X T that holds for any strongly continuous semigroup T generated by A T on a Banach space X , proved in [11, Proposition AII.3.8]). To apply this result first D(A∗T2 ) is determined, which results in the pairs (η, q) ∈ NBVm b ×R k such that for all component of η there exists νi ∈ NBVb with νi (1) = 0 satisfying  s νi (σ ) ηi (s) = dσ ∀s ∈ [0, 1]. 0 ci (σ ) Then the closure of D(A∗T2 ) is shown to be isometrically isomorphic to L 1 (0, 1)m × Rk where the norm in L 1 (0, 1)m is weighted by function c as stated, i.e. in the sense (2.3). T To represent X 2 2 ⊂ X 2∗ as L 1 (0, 1)m × Rk , we must define a pairing between L 1 (0, 1)m × Rk and X 2 . The natural pairing is the one defined by (ν, q), (v, r ) := φ((ν, q)), (v, r ) where φ is the isometric isomorphism between L 1 (0, 1)m × Rk and D(A∗T2 ) ⊂ NBVm b × R , which results in k     m  1 ν v νi (s) , = vi (s) ds + r · q. q r 0 ci (s) i=1 58 Page 8 of 33 C. Barril and À. Calsina J. Evol. Equ. T2 Finally it is proven that X 2 ∼ = X 2 . Indeed, using L 1 (0, 1)m × Rk as a represen- T2 T2 ∗ tation of X 2 clearly implies X 2 ∼ = L ∞ (0, 1)m × Rk . In this case the norm of L ∞ (0, 1)m is the standard one (i.e. is not affected by function c) if the pairing     m  1 v ν vi (s) , = νi (s)ds + r · q r q 0 ci (s) i=1 is used, as we do. On the other hand, the semigroup T2 on (L 1 (0, 1))m × Rk is specified by      ν ν̃(·; t, ν) T2 (t) = m  1 νi (s) q q + i=1 0 ci (s) 1− (ϕi (−t, s))ds i where i must be interpreted as a vector of Rk and where the ith component of ν̃(·; t, ν) is νi (ϕi (t, ·)) ν̃i (·, t, ν) = ci (·) ∂2 ϕi (t, ·)1+ (ϕi (−t, 1) − ·). ci (ϕi (t, ·)) Similarly, T2 is used to give an explicit formula for the semigroup T2∗ on L m ∞×R , k which results in the natural extension of T2 in L ∞ × R . m k Then, the infinitesimal generator of T2 on L 1 (0, 1)m × Rk is determined as  D(A T  ) = (ν, q) ∈ L 1 (0, 1)m × Rk | ν is absolutely continuous and ν(1) = 0 2 m and A T  (ν, q) = (cν  , i=1 νi (0) i ). By saying that ν is absolutely continuous we mean that the condition is satisfied component-wise, i.e. for each component of ν. The adjoint of A T  is consequently determined as 2  D(A∗T  ) = (v, r ) ∈ (L ∞ (0, 1))m × Rk | v is Lipschitz and v(0) = r (2.4) 2 and A∗T  (v, r ) = (−cv  , 0). By using [11, Proposition AII.3.8] one more time, we 2 T2 obtain X 2 as the closure of D(A∗T  ). When doing so we lose the Lipschitz condition 2 on v but the continuity remains. Therefore, T  X2 2 ∼ = (v, r ) ∈ C([0, 1], R)m × Rk | v(0) = r = X 2 , as desired.  Once shown that T is closed by ∗-integration on Y , we can use Theorem A2 to conclude that one and only one semiflow associated to (1.3) exists. This result is stated below in the form of a theorem. Theorem 1. Problem (1.1) is well posed on X = X 1 × X 2 defined in (1.5), i.e. there exists an open subset of [0, ∞) × X (in the induced topology) and a unique function from to X satisfying the following: J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 9 of 33 58 – for all initial condition x = (u 0 , v0 , r0 ) ∈ X there exists tx ∈ (0, ∞] such that [0, tx ) × {x} is the intersection of with [0, ∞) × {x}, – for all x = (u 0 , v0 , r0 ) ∈ X , function (·; x) from [0, tx ) to X is a mild solution of (1.1) (in the sense of Definition A3). Thus, satisfies the semigroup property, i.e. (t + s, x) = (t, (s, x)) for positive s and t with t + s < tx , whereas function (t; ·) is locally Lipschitz in x. 2.2. Linearisation around steady states Given a steady state (ū, v̄, r̄ ) ∈ X of the semiflow , Theorem A3 ensures that there exists a strongly continuous linear semigroup S(t) on X such that S(t) = D2 (t; (ū, v̄, r̄ )). Moreover, using that (v, r ) ∈ D(A∗T  ) if and only if v is Lips- 2 chitz and v(0) = r (see (2.4)), it follows that the domain of the generator A S of the semigroup S(t) (see Theorem A1) can be written as ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎫ ⎨ u 0 u ⎬ D(A S ) = ⎝ v ⎠ ∈ X | v is Lipschitz and ⎝ −c · v  ⎠ + DH(ū, v̄, r̄ ) ⎝ v ⎠ ∈ X ⎩ ⎭ r 0 r and then ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ u 0 u u A S ⎝ v ⎠ = ⎝ −c · v  ⎠ + DH(ū, v̄, r̄ ) ⎝ v ⎠ for all ⎝ v ⎠ ∈ D(A S ). (2.5) r 0 r r Next, in order to study the asymptotic behaviour of around (ū, v̄, r̄ ), we show that S(t) is eventually norm continuous (see Theorem 3), so that Theorem A7 can be applied and a characterization of the local dynamics around (ū, v̄, r̄ ) can be given in terms of the eigenvalues of the operator A S (see Theorem 4 at the end of the section). The proof is long and will occupy the rest of the section. Let us start noticing that DH(ū, v̄, r̄ ) has the form ⎛ ⎞ ⎛ ⎞ u B11 u + B12 v DH(ū, v̄, r̄ ) ⎝ v ⎠ = ⎝ B21 u + B22 v ⎠ . r K̃ (v, r ) where B11 ∈ L ∞ ((0, 1), Mn×n (R)), B12 ∈ L ∞ ((0, 1), Mn×m (R)), B21 ∈ L ∞ ((0, 1), Mm×n (R)), B22 ∈ L ∞ ((0, 1), Mm×m (R)), and K̃ is a bounded linear operator from X 2 into Rk (for instance, B11 (x) = D2 g(x, ū(x), v̄(x)). Thus, the generator A S can be formally written as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ u u u u AS ⎝ v ⎠ = A ⎝ v ⎠ + B ⎝ v ⎠ + K ⎝ v ⎠ , r r r r 58 Page 10 of 33 C. Barril and À. Calsina J. Evol. Equ. where B and K are bounded operators from X into X T ∗ . Their explicit expressions are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ u B11 u + B12 v u 0 B ⎝ v ⎠ = ⎝ B21 u + B22 v ⎠ and K ⎝v⎠ = ⎝ 0 ⎠. (2.6) r 0 r K̃ (v, r ) Notice that K is a compact operator because it takes values in a finite dimensional subspace of X ∗ , namely {0} × {0} × Rk . Thus, in order to show that S is eventually norm continuous it is enough to prove the eventually norm continuity of the simpler semigroup generated by A+ B. This is so because compact perturbations of eventually norm continuous semigroups are also eventually norm continuous (see [12, Proposition III.1.14] and Theorem A8 in the appendix). The proof of the eventual norm continuity of the semigroup S B generated by A + B is based on the series formula for S B given in Theorem A1. From that theorem we know that the series ∞  Sk , k=0 · with S0 = T and Sk = 0 T ∗ (· − s)B Sk−1 (s)ds for n > 1, converges uniformly (on compact time intervals) towards S B . By the Uniform Convergence Theorem, we know that if function Sk is continuous in [t0 , ∞) for each k, then so is S B . Therefore, it is enough to prove that each term in the series defining S B is continuous on [1, ∞) in order to conclude that S B is norm continuous from 1 onwards. The specific value at which the functions defining the series become continuous is 1 due to the fact that ci (x) ≥ 1 for all i, which implies that ϕi (−t, x) < 0 for all (t, x) ∈ (1, ∞) × [0, 1]. The proof consists in showing that the operator norm of (Sk (t + h) − Sk (t)) can be bounded as (Sk (t + h) − Sk (t)) < Mt,k |h| for all k ∈ N, t > 1 and |h| small enough, where Mt,k is a constant that only depends on t and k. In order to do that the image of a point (u, v, r ) by Sk (t + h) − Sk (t) is expressed as the sum of several terms and each of them is properly bounded. Hence the methodology is technical, though the tricks are mostly elementary. Unfortunately, we failed in giving a proof by induction on the index k. Next we are going to prove some intermediate steps that will lead to the result we desire. First, let us recall that the norms of X and j (X ) are equivalent (Proposition A.1), so that there exists M > 0 such that  ⎛ ⎞ ⎛ ⎞  ⎛ ⎞  u   u   u         j ⎝ v ⎠ ≤  v  ≤ M  j v ⎠  ⎝ ⎠  ⎝ . (2.7)     r   r   r  X ∗ X X ∗ J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 11 of 33 58 Then, for all k ≥ 1 and h ∈ (0, 1) observe that the definition of Sk implies  ⎛ ⎞  u    (Sk (t + h) − Sk (t)) ⎝ v ⎠    r  X  ⎛ ⎞ ⎛ ⎞   u u   t+h ∗  t ∗   M  0 T (t + h − s)B Sk−1 (s) ⎝ v ⎠ ds − 0 T (t − s)B Sk−1 (s) ⎝ v ⎠ ds    r r  (2.8) X ∗ ⎛ ⎞  û(·; t + h) − û(·; t)     M ⎝ ⎠  v̂(·; t + h) − v̂(·; t)   0  ∗ X  M( û(·; t + h) − û(·; t) L n∞ + v̂(·; t + h) − v̂(·; t) L m ∞ ) where, defining the projection πu : L n∞ × L m∞ × R → L ∞ so that πu (u, v, r ) = u, m n ⎛ ⎞  t u û(·; t) = πu B Sk−1 (s) v ⎠ ds, ⎝ (2.9) 0 r and, defining the projection πv : L n∞ × L m ∞ × R → L ∞ so that πv (u, v, r ) = v, m m ⎛ ⎞  t u v̂(·; t) = πv T ∗ (t − s)B Sk−1 (s) ⎝ v ⎠ ds. (2.10) 0 r From inequality (2.8) it follows that in order to obtain a bound of the form (Sk (t + h) − Sk (t)) < Mt h one has to focus on the terms û(·; t + h) − û(·; t) L n∞ and v̂(·; t + h) − v̂(·; t) L m∞ . This is what is done in the following lemmas. Lemma 1. For all t > 0 and h ∈ (0, 1), there exists M̃t > 0 such that the term û(·; t + h) − û(·; t) L n∞ defined in (2.8) can be bounded as û(·; t + h) − û(·; t) L n∞ < M̃t h (u, v, r ) X . Proof. Since the operator norm of Sk−1 (s) is uniformly bounded within bounded intervals of time, using (2.9) one has û(·; t + h) − û(·; t) L n∞  ⎛ ⎛ ⎞ ⎛ ⎞ ⎞   t+h u  t u      uπ ⎝ B Sk−1 (s) ⎝ v ⎠ ds − B Sk−1 (s) ⎝ v ⎠ ds ⎠   0 r 0 r  n L∞  ⎛ ⎞  ⎛ ⎞  t+h  u   u        B Sk−1 (s) ⎝ v ⎠ ds  h B Sk−1 (s)  ⎝ ⎠   sup  v  . (2.11) t  r  X ∗ s∈[0,t+1]  r  X  58 Page 12 of 33 C. Barril and À. Calsina J. Evol. Equ. Let us focus now on the terms v̂(·; t + h) − v̂(·; t) L m∞ of (2.8). Notice that B initially defined from X into X ∗ can be extended to a bounded operator from X ∗ into itself given by the same expression of B (see (2.6)). Thus, it is possible to enter the operators T ∗ (t − sk ) and B inside the integral that defines Sk (s) in (2.10). That is, v̂(·; t) can be written as ⎛ ⎞  t  sk u v̂(·; t) = πv T ∗ (t − sk )B T ∗ (sk − sk−1 )B Sk−2 (sk−1 ) ⎝ v ⎠ dsk−1 dsk 0 0 r ⎛ ⎞  t  sk u = πv T ∗ (t − sk )BT ∗ (sk − sk−1 )B Sk−2 (sk−1 ) ⎝ v ⎠ dsk−1 dsk , 0 0 r and, inductively,  t  sk  s2 v̂(·; t) = ··· 0 0 0 ⎛ ⎞ u πv T ∗ (t − sk )BT ∗ (sk − sk−1 )B · · · T ∗ (s2 − s1 )BT (s1 ) ⎝ v ⎠ ds1 · · · dsk−1 dsk . r (2.12) In order to simplify the above equation recall that X ∗ = L n∞ × L m ∞ × R and m ∗ ∗ ∗ observe that the operators T (sl+1 − sl )B : X → X for l ∈ {1, . . . , k} (setting sk+1 = t) can be synthesised as matrices of operators: ⎛ ⎞ B11 B12 0 T ∗ (sl+1 − sl )B ∼ ⎝ T̃ (sl+1 − sl )B21 T̃ (sl+1 − sl )B22 0⎠, 0 0 0 ∞ → L ∞ is defined as where T̃ (s) : L m m (T̃ (s)v)i = vi (ϕi (−s, ·))1+ (ϕi (−s, ·)) ∀i ∈ {1, . . . , m}. (2.13) The product of operators  1 T ∗ (sl+1 − sl )B = T ∗ (t − sk )BT ∗ (sk − sk−1 )B · · · T ∗ (s2 − s1 )B, l=k (with sk+1 = t) written as a matrix of operators, becomes: ⎛ ⎞ J11 J12 0 J ∼ ⎝ J21 J22 0⎠ 0 0 0 J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 13 of 33 58 where, for (i 0 , i k+1 ) ∈ {1, 2} × {1, 2},  2  2  2 Ji0 ik+1 = ··· Ji0 i1 i2 ···ik ik+1 (t − sk , sk − sk−1 , . . . , s2 − s1 ) i 1 =1 i 2 =1 i k =1 with Ji0 i1 i2 ···ik ik+1 (t − sk , sk − sk−1 , . . . , s2 − s1 ) = (T ∗ (t − sk )B)i0 i1 (T ∗ (sk − sk−1 )B)i1 i2 · · · (T ∗ (s2 − s1 )B)ik ik+1 . (2.14) In particular, using ṽ(·; s1 , v, r ) defined in (2.2) the integrand in (2.12) can be expressed as  1  ⎛ ⎞  u πv T (sl+1 − sl )B T (s1 ) v ⎠ = J21 u + J22 ṽ(·; s1 , v, r ) ∗ ⎝ l=k r so that (2.12) becomes the sum  2  2  2  2  2  2 v̂(·; t) = ··· I2 i1 i2 ···ik 1 (t, u) + ··· I2 i1 i2 ···ik 2 (t, v, r ) i 1 =1 i 2 =1 i k =1 i 1 =1 i 2 =1 i k =1 (2.15) where I2 i1 i2 ···ik 1 (t, u)  t  sk  s2 = ··· J2 i1 i2 ···ik 1 (t − sk , sk − sk−1 , . . . , s2 − s1 )uds1 · · · dsk−1 dsk 0 0 0 (2.16) and I2 i 1 i 2 ···i k 2 (t, v, r )  t  sk  s2 = ··· J2 i 1 i 2 ···i k 2 (t − sk , sk − sk−1 , . . . , s2 − s1 )ṽ(·; s1 , v, r )ds1 · · · dsk−1 dsk . 0 0 0 (2.17) Clearly, from (2.15) it follows v̂(·; t + h) − v̂(·; t) L m∞      ≤ i21 =1 i22 =1 · · · i2k =1  I2 i1 i2 ···ik 1 (t + h, u) − I2 i1 i2 ···ik 1 (t, u) L m (2.18)     ∞  + i21 =1 i22 =1 · · · i2k =1  I2 i1 i2 ···ik 2 (t + h, v, r ) − I2 i1 i2 ···ik 2 (t, v, r ) L m . ∞ In order to give a bound for v̂(·; t + h) − v̂(·; t) L m∞ , we show that each summand on the right hand side in (2.18) can be bounded properly, i.e. by something of the 58 Page 14 of 33 C. Barril and À. Calsina J. Evol. Equ. form h M̃t (u, v, r ) . This is done by performing a change of variables to the “shifted integrals” (the ones evaluated at t + h) so that the new integrand coincides with the integrand of the “unshifted integrals” (the ones evaluated at t). The equality between integrands makes, on the one hand, that the difference between integrals vanishes over the domain of integration that is common to both integrals. On the other hand, the domains of integration that are specific to each integral have a Lebesgue measure proportional to h. These statements are developed in the following. To do so we treat separately the terms in the first summation from those in the second summation. Lemma 2. Let I2 i1 i2 ···ik 1 (t, u) be defined as in (2.16). For all t > 0 and h ∈ (0, 1), there exists M̃t > 0 such that    I2 i i ···i 1 (t + h, u) − I2 i i ···i 1 (t, u) m < M̃t h (u, v, r ) X . 1 2 k 1 2 k L ∞ Proof. Applying a translation τh to the integration variables so that τh (s1 , s2 , . . . , sk ) = (s1 + h, s2 + h, . . . , sk + h) the integral I2 i1 i2 ···ik 1 (t + h, u) becomes  t  sk  s2 ··· J2 i1 i2 ···ik 1 (t − sk , sk − sk−1 , . . . , s2 − s1 )uds1 · · · dsk−1 dsk . −h −h −h Therefore, using T ∗ (s) X ∗ ≤ 1 for all s ≥ 0, one has    I2 i i ···i 1 (t + h, u) − I2 i i ···i 1 (t, u) m 1 2 k 1 2 k L∞  t  s  min{s2 ,0}  k =  ··· −h −h −h   t  J2 i1 i2 ···ik 1 (t − sk , sk − sk−1 , . . . , s2 − s1 )u ds1 · · · dsk−1 dsk   −h Lm ∞ ≤ h(t + h)k−1 B k (u, v, r ) ≤ h(t + 1)k−1 B k (u, v, r ) (2.19) as desired.  Let us consider now the terms in (2.18) of the form    I2 i i ···i 2 (t + h, v, r ) − I2 i i ···i 2 (t, v, r ) m . 1 2 k 1 2 k L ∞ In this case it is not clear whether it is possible to make a change of variables that transforms the integrand of I2 i1 i2 ···ik 2 (t +h, v, r ) into the integrand of I2 i1 i2 ···ik 2 (t, v, r ) as we did in the proof of the previous lemma. The problem is that the integration variables do not only appear in the function that determines J2 i1 i2 ···ik 2 , but they also play a role in the term ṽ. Apparently, finding a change of variables that transforms the integrand J2 i1 i2 ···ik 2 (t + h − sk , sk − sk−1 , . . . , s2 − s1 )ṽ(·; s1 , v, r ) J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 15 of 33 58 into J2 i1 i2 ···ik 2 (t − σk , σk − σk−1 , . . . , σ2 − σ1 )ṽ(·; σ1 , v, r ) amounts to solve the following system of k + 1 equations and k unknows: ⎧ ⎪ t + h − sk = t − σk ⎪ ⎪ s −s ⎪ ⎪ k−1 = σk − σk−1 ⎨ k .. , (2.20) ⎪ . ⎪ ⎪ ⎪ s2 − s1 = σ2 − σ1 ⎪ ⎩ s1 = σ1 which is impossible. However, the fact is that one of the equations in the system above is unnecessary to find the desired change of variables. To justify this let us distinguish the case I2 22···22 from the cases I2 i1 i2 ···ik 2 in which at least one index is 1. Lemma 3. Let I2 i1 i2 ···ik 2 (t, v, r ) be defined as in (2.17). Let il = 1 for some l ∈ {1, . . . , k}. For all t > 0 and h ∈ (0, 1), there exists M̃t > 0 such that    I2 i i ···i 2 (t + h, v, r ) − I2 i i ···i 2 (t, v, r ) m < M̃t h (u, v, r ) X . 1 2 k 1 2 k L ∞ Proof. Since il = 1, the term (T ∗ (sk − sk−1 )B)il il+1 appearing in (2.14) is (T ∗ (sk−l+1 − sk−l )B)il il+1 = (T ∗ (sk−l+1 − sk−l )B)1il+1 = B1il+1 , so that J2 i1 i2 ···ik 2 is independent of the difference sk−l+1 − sk−l . Thus, the desired change of variables can be obtained without imposing the equality sk−l+1 − sk−l = σk−l+1 − σk−l . This means that this equation can be removed from system (2.20) so that it becomes compatible. The solution of the reduced system is the desired change of variables, which is a translation τh given by τh (s1 , s2 , . . . , sk ) = (s1 , . . . , sk−l , sk−l+1 + h . . . , sk + h). By using this transformation on the “shifted integral” one has I2 i1 i2 ···ik 2 (t + h, v, r )  t  sk  sk−l+2  sk−l+1 +h  sk−l  s2 = ··· ··· −h −h −h 0 0 0 J2 i1 i2 ···ik 2 (t − sk , . . . , s2 − s1 )ṽ(·; s1 , v, r )ds1 · · · dsk , and, similarly as done in (2.19), one concludes    I2 i i ···i 2 (t + h, v, r ) − I2 i i ···i 2 (t, v, r ) m 1 2 k 1 2 k L ∞ ≤ h(t + 1)k−1 B k (u, v, r ) . (2.21)  58 Page 16 of 33 C. Barril and À. Calsina J. Evol. Equ. The previous proof cannot be applied to give an analogous bound for the terms of the form I222···22 (t + h, v, r ) − I222···22 (t, v, r ) Lm ∞ . The problem is that, in this case, the term J222···22 depends on all the differences sl+1 − sl with l ∈ {1, . . . k}, so that all the first k rows of system (2.20) have to be imposed. The bound, however, can be proven by showing that the last equation of (2.20) is not needed to give a suitable change of variables. Lemma 4. Let di = si+1 − si for i ∈ {1, . . . , k} and sk+1 = t. If t > 1, the functions 1+ (ϕl1 (−s1 , ·) ◦ ϕl2 (−d1 , ·) ◦ · · · ◦ ϕlk (−dk−1 , ·) ◦ ϕi (−dk , ·)) and 1− (ϕl1 (−s1 , ·) ◦ ϕl2 (−d1 , ·) ◦ · · · ◦ ϕlk (−dk−1 , ·) ◦ ϕi (−dk , ·)) seen as elements of L ∞ (0, 1) are the constant functions 0 and 1 respectively. Proof. Notice that for all x ∈ [0, 1] and t > 1 one has ϕl1 (−s1 , ·) ◦ ϕl2 (−d1 , ·) ◦ · · · ◦ ϕlk (−dk−1 , ·) ◦ ϕi (−dk , ·) < 0. Indeed, using that ∂t ϕ j (t, x) ≥ 1 for all j ∈ {1, . . . , m}, which implies ϕ j (−d, x) ≤ ϕ j (0, x) − d = x − d for all d > 0, we deduce ϕl1 (−s1 , ·) ◦ ϕl2 (−d1 , ·) ◦ · · · ◦ ϕlk (−dk−1 , ·) ◦ ϕi (−dk , ·)(x) ≤ ϕl2 (−d1 , ·) ◦ · · · ◦ ϕlk (−dk−1 , ·) ◦ ϕi (−dk , ·)(x) − s1 ≤ ϕl3 (−d2 , ·) ◦ · · · ◦ ϕlk (−dk−1 , ·) ◦ ϕi (−dk , ·)(x) − d1 − s1 ≤ · · · ≤ k−2 ≤ ϕlk (−dk−1 , ·) ◦ ϕi (−dk , ·)(x) − i=1 d − s1 k−1 k i ≤ ϕi (−dk , x) − i=1 di − s1 ≤ x − i=1 di − s1 k = x − i=1 (si+1 − si ) − s1 = x − t < 0 since t > 1 and x ∈ [0, 1].  Lemma 5. Let J222···22 (t − sk , sk − sk−1 , . . . , s2 − s1 ) be defined as in (2.14). For all t > 1 the following holds J222···22 (t − sk , sk − sk−1 , . . . , s2 − s1 )ṽ(·; s1 , v, r ) = J222···22 (t − sk , sk − sk−1 , . . . , s2 − s1 )r. (2.22) Proof. Recall the definition of T̃ given in (2.13). Denoting dl = sl+1 − sl , the ith component of J222···22 (dk , . . . , d1 )v is (J222···22 (dk , . . . , d1 )v)i = (T̃ (dk )B22 T̃ (dk−1 )B22 · · · T̃ (d1 )B22 v)i m  m = ··· T̃i (dk )B22,ilk T̃lk (dk−1 )B22,lk lk−1 · · · T̃l2 (d1 )(B22,l2 l1 vl1 ) lk =1 l1 =1  m  m = ··· (T̃i (dk )B22,ilk T̃lk (dk−1 )B22,lk lk−1 · · · T̃l2 (d1 )B22,l2 l1 ) lk =1 l1 =1 (T̃i (dk )T̃lk (dk−1 ) · · · T̃l2 (d1 )vl1 ). (2.23) J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 17 of 33 58 In the last equality we have used that, for all i ∈ {1, . . . , m} and for all f, g ∈ L ∞ (0, 1), the operator T̃i (d) satisfies T̃i (d)( f g) = f (ϕi (−d, ·))g(ϕi (−d, ·))1+ (ϕi (−d, ·)) = ( f (ϕi (−d, ·))1+ (ϕi (−d, ·)))(g(ϕi (−d, ·))1+ (ϕi (−d, ·))) = (T̃i (d) f )(T̃i (d)g), (2.24) so that for all triad b, f, g ∈ L ∞ (0, 1) (for instance b = B22,l3 l2 , f = B22,l2 l1 and g = vl1 ) one has b T̃i (d)( f g) = (b T̃i (d) f )(T̃i (d)g). In particular, the ith component of J222···22 (dk , . . . , d1 )ṽ(·; s1 , v, r ) is (J222···22 (dk , . . . , d1 )ṽ(·; s1 , v, r ))i = SUMv + SUMr with m m SUMv = lk =1 · · · l1 =1 (T̃i (dk )B22,ilk T̃lk (dk−1 )B22,lk lk−1· · · T̃l2 (d1 )B22,l2 l1 ) (T̃i (dk )T̃lk (dk−1 ) · · · T̃l2 (d1 )(vl1 (ϕl1 (−s1 , ·))1+ (ϕl1 (−s1 , ·)))) and m m SUMr = lk =1 · · · l1 =1 (T̃i (dk )B22,ilk T̃lk (dk−1 )B22,lk lk−1 · · · T̃l2 (d1 )B22,l2 l1 ) (T̃i (dk )T̃lk (dk−1 ) · · · T̃l2 (d1 )(l1 r 1− (ϕl1 (−s1 , ·)))). Since the operators T̃i (d) satisfy T̃i (d)( f g) = f (ϕi (−d, ·))g(ϕi (−d, ·))1+ (ϕi (−d, ·)) = (T̃i (d) f )g(ϕi (−d, ·)), for all f ∈ L ∞ (0, 1) and g ∈ L ∞ (R) (and agreeing that the product of a function f in L ∞ (0, 1) with a function g in L ∞ (R) is the product of f with the projection of g in L ∞ (0, 1)), the factors in each summand of SUMv that depend on v, i.e. the terms of the form T̃i (dk )T̃lk (dk−1 ) · · · T̃l2 (d1 )(vl1 (ϕl1 (−s1 , ·))1+ (ϕl1 (−s1 , ·))), can be written as the product (T̃i (dk )T̃lk (dk−1 ) · · · T̃l2 (d1 )vl1 (ϕl1 (−s1 , ·))) 1+ (ϕl1 (−s1 , ·) ◦ ϕl2 (−d1 , ·) ◦ · · · ◦ ϕlk (−dk−1 , ·) ◦ ϕi (−dk , ·)), whereas the factors in each summand of SUMr that depend on r can be written as (T̃i (dk )T̃lk (dk−1 ) · · · T̃l2 (d1 )l1 r ) 1− (ϕl1 (−s1 , ·) ◦ ϕl2 (−d1 , ·) ◦ · · · ◦ ϕlk (−dk−1 , ·) ◦ ϕi (−dk , ·)). 58 Page 18 of 33 C. Barril and À. Calsina J. Evol. Equ. Now, by Lemma (4) we conclude that SUMv is zero whereas SUMr equals m m lk =1 · · · l1 =1 (T̃i (dk )B22,ilk T̃lk (dk−1 )B22,lk lk−1 · · · T̃l2 (d1 )B22,l2 l1 ) (T̃i (dk )T̃lk (dk−1 ) · · · T̃l2 (d1 )l1 r ), which clearly coincides with the ith component of J222···22 (dk , . . . , d1 )r if one uses the expression (2.23). That is, for all i ∈ {1, . . . , m} one has (J222···22 (dk , . . . , d1 )ṽ(·; s1 , v, r ))i = (J222···22 (dk , . . . , d1 )r )i , which implies the equality stated in (2.22).  Lemma 6. Let I222···22 (t, v, r ) be defined as in (2.17). For all t > 1 and h ∈ (0, 1), there exists M̃t > 0 such that I222···22 (t + h, v, r ) − I222···22 (t, v, r ) Lm ∞ < M̃t h (u, v, r ) X . Proof. Lemma 5 implies that, if t > 1, function ṽ in the integrand of I222···22 (t, v, r ) can be replaced by the constant r . Then, the solution of system (2.20) with the last equation removed gives a change of variables, namely τh (s1 , s2 , . . . , sk ) = (s1 + h, s2 + h, . . . , sk + h), such that the integral I222···22 (t + h, v, r ) transforms into  t  sk  s2 ··· J222···22 (t − sk , sk − sk−1 , . . . , s2 − s1 )r ds1 · · · dsk−1 dsk . −h −h −h Finally, similarly as it is done in (2.19) and taking into account r = v(0), one concludes I222···22 (t + h, v, r ) − I222···22 (t, v, r ) Lm ∞ ≤ h(t + 1) k−1 B k (u, v, r ) . (2.25)  Lemma 7. For all t > 1 and h ∈ (0, 1), there exists M̃t > 0 such that the term v̂(·; t + h) − v̂(·; t) L m∞ defined in (2.8) can be bounded as v̂(·; t + h) − v̂(·; t) Lm ∞ ≤ M̃t h (u, v, r ) X . Proof. By using the bounds (2.19), (2.21) and (2.25) in inequality (2.18) it follows ⎛ ⎞  u    k ⎝ ⎠ v̂(·; t + h) − v̂(·; t) Lm ∞ ≤ 2k+1 h(t + 1)k−1 B  v  . (2.26)  r  X  J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 19 of 33 58 Theorem 2. Let T and T ∗ be defined by (2.1) on X and X T ∗ respectively. Let B be defined by (2.6). The functions S0 (t) = T (t), t Sk (t) = j −1 0 T ∗ (t − s)B Sk−1 (s)ds, ∀k ∈ N from [0, ∞) to L(X ) (the Banach space of the bounded linear operators on X with the operator norm) are continuous on [1, ∞). Proof. The case k = 0 follows immediately from the definition of T (·) in ( 2.1). Indeed, let t > 1 and h > 1 − t. Then  ⎛ ⎞ ⎛ ⎞  u       u−u  (T (t + h) − T (t)) ⎝ v ⎠ = ⎝ r − r ⎠ = 0,      r   r −r  that is T (t + h) − T (t) = 0 for all h > 1 − t, and in particular the limit as h tends to zero is also zero. The general case k > 0 is a consequences of the previous lemmas. If h ∈ (0, 1) and t > 0, then Lemmas 1 and 7 applied to inequality (2.8) yield (Sk (t + h) − Sk (t)) ≤ h Mt (2.27) with Mt = M( B sup Sk−1 (s) + 2k+1 (t + 1)k−1 B k ). s∈[0,t+1] If t > 1 and h ∈ (max{1 − t, −1}, 0), then inequality (2.27) can be applied as (Sk (t + h) − Sk (t)) = (Sk (t + h + |h|) − Sk (t + h)) ≤ kt+h |h| ≤ Mt |h|, since, on the one hand, t + h is still bigger than 1 and, on the other hand, the constant Mt is increasing as a function of t. By combining the results for positive and negative h, one finally concludes that for t > 1 and h ∈ (max{1 − t, −1}, 1) there exists a constant Mt (which depends on t) such that Sk (t + h) − Sk (t) X ≤ |h|Mt .  As a corollary of Theorem 2 and the uniform convergence of the series defining S B , it follows that Theorem 3. The semigroup S B generated by A + B (in the sense of Theorem A1) is eventually norm continuous. Moreover, since K is a compact operator from X into X ∗ , the semigroup S generated by A + B + K (in the sense of Theorem A1, using either A + (B + K ) or (A + B) + K ) is eventually norm continuous (as an application of Theorem A8). 58 Page 20 of 33 C. Barril and À. Calsina J. Evol. Equ. Finally, the eventual norm continuity of the linearised semigroup S makes possible to apply Theorem A7, so that the following result on linearisation specific to problem (1.1) can be stated: Theorem 4. Let (ū, v̄, r̄ ) ∈ X be a steady state of the semiflow associated to (1.1) and let A S be the operator defined in (2.5). Then, (i) (ū, v̄, r̄ ) is locally asymptotically stable if s(A S ) < 0, (ii) (ū, v̄, r̄ ) is unstable if there exists ω > 0 such that the spectrum of A S within the region {λ ∈ C | Re(λ) > ω} is non-empty and is composed only by a finite number of eigenvalues with finite algebraic multiplicity. 3. Applications and discussion The system presented in Sect. 2 was originally inspired by an epidemiological problem involving pathogens spreading through the oral-fecal route [2,4] (see also [1,14]). Specifically, in [2] we analysed a biological system involving n hosts (animals) and m species of microorganisms living in their intestines, either as free particles in the lumen or attached to the epithelial wall. By calling u h,s (x, t) and vh,s (x, t) the densities of attached and luminal microbes of type s in the host h respectively, and rs (t) the density of microbes s in the soil, all of them at time t, the dynamical equations are ⎧ ⎨ ∂t u h,s (t, x) = gh,s (x, u h (t, x), vh (t, x)), ∂t vh,s (t, x) = −∂x (ch (x)vh,s (t, x)) + f h,s (x, u h (t, x), vh (t, x)), (3.1) ⎩ drs (t)   dt = m s (r (t)) + h∈H c h (l h )v h,s (t, l h ) − h∈H λ h,s r s (t), with h ∈ {1, . . . , n} =: H and s ∈ {1, . . . , m} =: S. In the above equations the notation u h = (u h,1 , . . . , u h,m ), vh = (vh,1 , . . . , vh,m ) and r = (r1 , . . . , rm ) is used. The parameter lh is the intestine length of host h (thus, the spatial domain of both u h and vh is [0, lh ]) and ch (x) stands for the velocity of its intestinal flow (notice that it is implicitly assumed that advection dominates diffusion). The functions gh,s and f h,s take into account the ecological processes happening locally at the position x of the intestine. Besides replication and mortality of bacteria, these functions may also reflect migration between epithelium and lumen, competition interactions or whatever we are interested in. Similarly, the function m s describes the ecology in the external media of the population of type s. Finally, we assume that microbes enter the intestine in a rate which is proportional to their amount in the soil. Thus, λh,s represents a kind of ingestion rate of particles of type s by host h. Consequently, a boundary condition for vh,s must be incorporated relating such reinfection term, which is ch (0)vh,s (0, t) = λh,s rs ∀(h, s) ∈ H × S. (3.2) The diversity in gut lengths across hosts makes that, a priori, system (3.1) has not the form of system (1.1). Fortunately, we can perform a change in the spatial variables J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 21 of 33 58 to rewrite (3.1) properly in the form of a system as (1.1). Such a change is possible because microbes within a host do not affect what happens in a different host. Indeed, by defining ṽh (x, t) := vh (lh x, t) and ũ h (x, t) := u h (lh x, t), we have, on the one hand, that the spatial domain of ũ h,s and ṽh,s is the interval [0, 1] for all (h, s) ∈ H × S and, on the other hand, that system (3.1) transforms into ⎧ ⎨ ∂t ũ h,s (t, x) = gh,s (lh x, ũ h (t, x), ṽh (t, x)), ∂t ṽh,s (t, x) = −∂x (ch (lh x)ṽh,s (t, x)) + f h,s (lh x, ũ h (t, x), ṽh (t, x)), (3.3) ⎩ drs (t)   dt = m s (r (t)) + h∈H ch (l h )ṽh,s (t, 1) − h∈H λh,s rs (t). with the boundary condition λh,s ṽh,s (0, t) = rs (t) ∀(h, s) ∈ H × S. ch (0) In particular, by considering vectors ũ and ṽ to be indexed by one number instead of two, for example by writing ũ m(h−1)+s instead of ũ h,s (analogously for ṽ), then the initial value problem associated to (3.3) has the form of problem (1.1). Thus, in order to apply Theorems 1 and 4 to system (3.3) the phase space we have to work on is the Banach space X = X1 × X2 with X 1 = L ∞ (0, 1)n×m and   λh,s X 2 = (ṽ, r ) ∈ C([0, 1], R)n×m × Rm | ṽm(h−1)+s (0) = rs ∀(h, s) ∈ H × S . ch (0) In [2] problem (3.1) was analysed formally more than theoretically. In the present paper we have proved that Theorems 1 and 4 apply to problem (3.3) (which is equivalent to system (3.1)), so that, as a corollary, we can state that the procedures followed in [2] are well supported by the theory of dual semigroups. In addition, Theorem 4 also implies that the conjecture used in [2] about the asymptotic dynamics of the system around steady states is true. To illustrate the theory developed in this paper let us consider the simplest case of the ones treated in [2]: problem (3.1) and (3.2) with a single host, a single bacterial strain and assuming that the intestinal flow velocity is constant at all points (i.e. that c(x) is constant). In this case, and using g(u, v, x) = uγ1 (u) − αu + δv, f (u, v, x) = vγ2 (v) + αu − δv and m(r ) = −μr the equations reduce to ⎧ ⎪ ∂t u(t, x) = u(t, x)γ1 (u(t, x)) − αu(t, x) + δv(t, x), ⎪ ⎨ ∂ v(t, x) = −c∂ (v(t, x)) + v(t, x)γ (v(t, x)) + αu(t, x) − δv(t, x), t x 2 dr (t) (3.4) ⎪ ⎪ = −μr (t) + cv(t, l) − λr (t), ⎩ dt cv(t, 0) = λr (t). Here α and δ represent attachment and detachment rates (to and from the intestinal epithelium), μ is a mortality rate of the bacteria found in the environment (outside 58 Page 22 of 33 C. Barril and À. Calsina J. Evol. Equ. the host) and γ1 and γ2 are per capita reproduction rates in the epithelium and in the lumen of the intestine. In [2] it is shown that if γ1 and γ2 are decreasing and negative for large arguments, then (3.4) has at most one positive stationary solution. Such a positive equilibrium exists if and only if • γ1 (0) ≥ δ, or • γ1 (0) < δ and   λ l γ2 (0)−α+ δ−γαδ (0) ec 1 > 1. λ+μ The local behaviour of this equilibria, denoted by (ū(·), v̄(·), r̄ ), can be determined applying Theorem 4. To do this first we specify the generator A S of the linearised semigroup around this equilibria (see (2.5) and also (1.4)), which is ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ u a1 (·)u(·) + αv(·) u A S ⎝ v ⎠ = ⎝ −c · v  (·) + a2 (·)v(·) + δu(·) ⎠ for all ⎝ v ⎠ ∈ D(A S ), r cv(l) − λr − μr r where a1 (·) = γ1 (ū(·)) − δ + γ1 (ū(·))ū(·), . a2 (·) = γ2 (v̄(·)) − α + γ2 (v̄(·))v̄(·). Then the spectral values of A S are determined by studying which operators A S − ηId defined on D(A S ) fail to have a continuum inverse. This leads to the characteristic equation whose solutions are the eigenvalues of A S and which can be used to give the spectral bound of A S (see Lemma 3.5 in [2] for more details). For this particular example it turns out that the positive stationary solution, when does exist, is always locally asymptotically stable. Beyond the model treated in [2], problem (1.3) can also be applied to determine if coexistence scenarios exist in the chemostat model of [1,14] and, in general, to any system of living beings inhabiting networks where advection is much more intense than diffusion. In the particular case of gut microogranisms, however, the coexistence of different bacteria has implications not only on the microbial ecology, but also on the host development. Indeed, physiological diversity between microbes may create dis- tinguished biofilm patterns along the intestine which, in turn, could play a role during tissue differentiation. A “pathogenic” pattern would result in intestines more vulnera- ble to infections or in microbiomes unable to supply essential nutrients. In this sense, improving our understanding about stable gastro-intestinal ecosystems will be help- ful for therapists dedicated to reshape the microbiome of patients with dysfunctional microbial patterns. In addition to the biological implications of the model proposed above, the present work has motivated some mathematical questions. Recall that in Sect. 3 we proved that a certain semigroup was eventually norm continuous. This property implied the Spectral Mapping Theorem, which essentially gives a one to one relation between the J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 23 of 33 58 spectrum of the semigroup evaluated at t and the spectrum of its generator. The proof, however, relied on the particular structure of the system we considered and, from our point of view, this was not optimal. By studying alternative ways to treat the problem, we realised that the kind of semigroup we were analysing was obtained by perturbing an eventually continuous semigroup by a bounded operator (in fact, the bounded operator was atypical in the sense that it was defined from the base space X to X T ∗ , with T being the unperturbed semigroup, but let us set aside this particularity now). A natural question at this point is whether the perturbation of an eventually continuous semigroup by a bounded perturbation is also eventually norm continuous. As Nagel and Engel show in their book, this is not true in general (see [12, Example III.1.15], in which the unperturbed semigroup is even nilpotent), although if the bounded perturbation is also compact, then the perturbed semigroup does inherit the eventual continuity of the unperturbed semigroup (see Theorem A8). Since in our case the compactness of the perturbation was not something we could assume, deducing the eventual continuity of the perturbed semigroup by means of general results seemed unfeasible. However, our final goal was not to show that the perturbed semigroup was eventually continuous, we wanted to prove the Spectral Mapping Theorem for the perturbed semigroup, which is something weaker. In this sense, we would need a general result as the following. Conjecture 1. Let X be a Banach space. Let T be an eventually norm continuous semigroup on X generated by A. Let B be a bounded operator on X . Then the Spectral Mapping Theorem holds for the semigroup S generated by A + B, i.e. σ (S(t)) \ {0} = et σ (A+B) holds for all t ≥ 0. Similarly, a weaker version of the above conjecture that, if proved to be true, would also imply Theorem 4 as a corollary is the following. Conjecture 2. Let X be a Banach space. Let T be an eventually norm continuous semigroup on X generated by A. Let B be a bounded operator on X . Then the growth bound of the semigroup S generated by A + B is equal to the spectral bound of the operator A + B, i.e. ω0 (S) = s(A + B). Our attempts to prove or disprove any of these two conjectures were unfruitful, but since we have not found any reference to these questions in the specialised literature [12,26], we present it here for those with more expertise. Acknowledgements Both authors were partially supported by Spanish research projects MTM2014-52402- C03-2P and MTM2017-84214-C2-2-P. C. Barril was also partially supported by Span- ish Ministry of Education Grant FPU13/04333. Funding Open Access Funding provided by Universitat Autonoma de Barcelona. 58 Page 24 of 33 C. Barril and À. Calsina J. Evol. Equ. Data availability Data sharing not applicable to this article as no datasets were gen- erated or analysed during the current study. Open Access. 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Appendix A Sun-dual formalism and evolution equations In this appendix an introduction of the work done by Clement et al. [6,7] on sun- dual semigroups is given. Although similar brief summaries exist written by the same authors [8], we include our own in order to present a more comprehensible work and because slight variations are introduced to deal with non-sun-reflexive spaces for which the variation of constants formula is well defined (see [9] for another reference where this semilinear formulation is undertaken without the sun-reflexivity hypothesis). Thus some of the theorems stated below are not exactly the same as the ones found in [6,7], even though the arguments to prove them can be applied essentially in the same way. A.1 Linear theory Let X be a Banach space, and denote by X ∗ its dual space. An element x ∗ ∈ X ∗ is, by definition, a linear continuous operator from X to R. We denote the image of an element x ∈ X by x ∗ with the bracket x ∗ , x . Given a closed operator C on X , its adjoint operator C ∗ is a linear operator from X to X ∗ with domain ∗ D(C ∗ ) = {x ∗ ∈ X ∗ | ∃ζ ∗ ∈ X ∗ such that x ∗ , C x = ζ ∗ , x ∀x ∈ D(C) ⊂ X }. It turns out that if x ∗ ∈ D(C ∗ ) then only one ζ ∗ ∈ X ∗ exists satisfying x ∗ , C x = ζ ∗ , x for all x ∈ D(C), so that the image of x ∗ by C ∗ is defined unequivocally as C ∗ x ∗ = ζ ∗ . In particular if C is a bounded operator, then C ∗ is also a bounded operator and satisfies C ∗ x ∗ , x = x ∗ , C x for all x ∗ ∈ X ∗ and x ∈ X . Given a strongly continuous semigroup T on X , the sun-dual space of X relative to T is a subspace of the dual space X ∗ defined by: ! X T := x ∗ ∈ X ∗ | T ∗ (t)x ∗ − x ∗ → 0 as t ↓ 0 , J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 25 of 33 58 where T ∗ (t) is the adjoint of the operator T (t). We denote by T  (t) the restriction of T ∗ (t) to X T , so that T  is, by construction, a strongly continuous semigroup on X T whose infinitesimal generator is denoted as A T  . Repeating this procedure on the pair (T  , X T ) we define the double-sun-dual of X relative to T as: X T := (X T )T  . The canonical injection j : X −→X T ∗ := (X T )∗ is determined by the pairing j (x), x  = x  , x ∀x  ∈ X T . In [6] it is shown that j (x) X T ∗ ≤ x X ≤ M j (x) X T ∗ , (A.1) where M is a constant which depends on T . Among other things, this implies that j (X ) ⊂ X T (A.2) since it is easily checked that T ∗ (t) j (x) = j T (t)x and then T ∗ (t) j (x) − j (x) X T ∗ ≤ T (t)x − x X −→ 0 as t ↓ 0. The canonical injection makes possible to introduce two important concepts for the development of the theory. Definition A1. X is said to be sun-reflexive relative to a strongly continuous semi- group T on X if j (X ) = X T . Definition A2. Let T be a strongly continuous semigroup on X . Let Y be a subspace of X T ∗ . We say that T is closed by ∗-integration on Y if, for all f ∈ C([0, ∞), Y ) and for all t ≥ 0,  t T ∗ (t − s) f (s)ds ∈ j (X ). (A.3) 0 The integral in (A.3) must be understood as an element of X T ∗ , and specifically (due to Bochner integral properties) as the functional satisfying  t   t ∗  T (t − s) f (s)ds, x = T ∗ (t − s) f (s), x  ds. 0 0 Notice that a semigroup T is always closed by ∗-integration on j (X ) since  t  t T ∗ (t − s) f (s)ds = j T (t − s) j −1 f (s)ds. 0 0 A well-known result of the theory states that the kind of integrals given by (A.3) take values not in the whole space X T ∗ but in the subset X T . 58 Page 26 of 33 C. Barril and À. Calsina J. Evol. Equ. Proposition A1 ([6, Theorem 3.2]). Let T be a strongly continuous semigroup on X and f ∈ C([0, ∞), X T ∗ ). Then  t T ∗ (t − s) f (s) ∈ X T ∀t ≥ 0. 0 Taking into account this proposition together with inclusion (A.2) it follows: Corollary A1. If X is sun-reflexive relative to T , then T is closed by ∗-integration on X T ∗ . As already commented, most propositions in the series of papers [6,7] as well as in the book [11] assume that a given Banach space X is sun-reflexive relative to a semigroup T . However, it is possible to prove similar results by means of analogous arguments assuming the closedness of T by ∗-integration on a subspace Y of X T ∗ . Let us give the reformulated statements we need for the present thesis. Theorem A1 ([6, Theorem 4.2]). Let T be a strongly continuous semigroup generated by A and closed by ∗-integration on Y ⊂ X T ∗ . Let B be a bounded linear operator from X into Y . Then the equation  t S(t)x = T (t)x + j −1 T ∗ (t − s)B S(s)xds (A.4) 0 uniquely defines a strongly continuous semigroup S on X . The partial sums of ∞  Sk , k=0 with S0 = T and  t −1 Sk+1 (t) = j T ∗ (t − s)B Sk (s)ds ∀t ≥ 0, 0 converge towards S uniformly on compact intervals, i.e.  n lim sup S(t) − Sk (t) = 0 ∀τ > 0. n→∞ t∈[0,τ ] k=0 The generator of S is A S with domain D(A S ) = {x ∈ X | j (x) ∈ D(A∗T  ) and A∗T  j (x) + Bx ∈ j (X )} and images A S x = j −1 (A∗T  j (x) + Bx). The semigroup implicitly defined by equation (A.4) makes natural to ask for its sun- dual spaces, that is for X  S and X  S . The following result shows that these spaces are determined just by the generator A, so they do not depend on the perturbation B. J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 27 of 33 58 Proposition A2 ([6, Lemma 4.3]). Let T , A, B and S be as in Theorem A1. Then X  S = X T and X  S = X T . Not only the sun-dual spaces are invariant with respect bounded perturbations from X to Y . The property of being closed by ∗-integration on Y is also satisfied by the perturbed semigroup S. Notice that this makes sense since Proposition A2 ensures that Y is a subspace of X  S ∗ . This is stated formally in the following proposition. The same proposition relates the evolution family obtained when T is perturbed by B + f with the evolution family obtained when S is perturbed by f , where f is a continuous function of time from [0, ∞) to Y . Proposition A3 ([7, Proposition 2.5]). Let T , A, B and S be as in Theorem A1. Then S is closed by ∗-integration on Y . Moreover, for every x ∈ X and every function f ∈ C([0, ∞), Y ), u(t; x) defined as  t u(t; x) = S(t)x + j −1 S ∗ (t − τ ) f (τ )dτ 0 is the only solution of  t −1 u(t; x) = T (t)x + j T ∗ (t − τ )(Bu(τ ; x) + f (τ ))dτ. 0 Corollary A2. Let T , A, B and S be as in Theorem A1. Let B1 and B2 be bounded operators from X into Y ⊂ X T ∗ such that B = B1 + B2 , and let S1 be the semigroup obtained when T is perturbed by B1 . Then S is equal to the semigroup obtained when S1 is perturbed by B2 (in the sense of Theorem A1). The above reformulation of the perturbation theory for dual semigroups is useful when, on the one hand, the unperturbed semigroup T is defined on a non-sun-reflexive space X , but, on the other hand, T is closed by ∗-integration on some subspace Y bigger than j (X ) (so that the standard semilinear formulation is not enough to solve the problem). At this point it is mandatory to show that semigroups satisfying these properties do exist. Example A1. Let X 1 and X 2 be Banach spaces. Consider X = X 1 × X 2 with the usual norm, and let T = diag(T1 , T2 ) be a strongly continuous semigroup with a “diagonal” structure, i.e. T (t)(x1 , x2 ) = (T1 (t)x1 , T2 (t)x2 ) for all t ≥ 0. Let X 2 be sun-reflexive T ∗ with respect T2 . For i ∈ {1, 2} let ji be the canonical inclusion from X i into X i i T2 ∗ and define Y = j1 (X 1 ) × X 2 . Then T is closed by ∗-integration on Y . Proof. Take f ∈ C([0, ∞), Y ) arbitrary, and define f 1 and f 2 the component func- T ∗ tions of f in j1 (X 1 ) and X 2 2 respectively. On the one hand, since j1 is a linear bounded operator and j1 T1 (t) = T1∗ (t) j1 , then  t  t  ∗ −1 T1 (t − s) f 1 (s)ds = j1 T1 (t − s) j1 f 1 (s)ds ∈ j1 (X 1 ). 0 0 58 Page 28 of 33 C. Barril and À. Calsina J. Evol. Equ. T2 ∗ On the other hand, since f 2 is norm continuous from [0, ∞) to X 2 , Proposition A1 implies  t T T2∗ (t − s) f 2 (s)ds ∈ X 2 2 , 0 T2 and this is enough because X 2 = j2 (X 2 ) due to the sun-reflexivity condition.  Notice that the subspace Y in the previous example is in general bigger than j (X ) T ∗ T because X 2 2 is in general bigger than X 2 2 . Notice also that X could be non-sun- reflexive if, for example, T1 (t) = Id for all t ≥ 0 and X 1 were non-reflexive. Arguably one could say that the example above is very degenerate due to the diagonal structure of the semigroup T . However, Proposition A3 allows us to take any perturbation of T by a bounded linear perturbation from X to Y , which give a collection of less trivial examples. A.2. Semi-linear theory Consider the initial value problem ⎧ dv(t) ⎨ dt = A T v(t) + j −1 (H(v(t))) , (A.5) ⎩ v(0) = x ∈ X where A T is the generator of a strongly continuous semigroup T which is closed by ∗-integration on Y (see Definition A2) and H : X → Y ⊂ X T ∗ is a Lipschitz function. Since H takes values in Y ⊂ X T ∗ , system (A.5) is a non-standard initial value problem which even fails to be well defined because in general j (X )  Y (notice that a classical solution v(·) of (A.5) must satisfy H(v(t)) ∈ j (X ) for all t ≥ 0). However, such a problem admits a generalised version of the variation of constants equation, which takes the form  t  v(t) = T (t)x + j −1 T ∗ (t − s)H(v(s))ds , (A.6) 0 thanks to the fact that T is closed by ∗-integration on Y . The functions v : [0, ∞) → X satisfying this integral equation are their solutions. A classical solution of (A.5) (if any) is a solution of (A.6), which motivates the notion of mild solution: Definition A3. A function v : [0, ∞) → X is a mild solution of (A.5) if it is contin- uous and it satisfies the integral equation (A.6). Two fundamental properties of mild solutions are given in the following theorem. Theorem A2 ([7, Theorem 3.1]). For every x ∈ X there exists a unique mild solution v(·; x) of (A.5). Moreover, v satisfies the semigroup property v(t+s; x) = v(t; v(s; x)) and v(t; ·) is Lipschitz. We refer to v(·; ·) as the semiflow of (A.5). J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 29 of 33 58 When H is Fréchet differentiable, a result on the differentiability with respect to initial conditions around a steady state can be given. Here, by steady state we mean an element x̄ ∈ X such that  t  x̄ = T (t)x̄ + j −1 T ∗ (t − s)H(x̄)ds ∀t ≥ 0. 0 As it occurs in the standard semilinear formulation, the steady states satisfying the above integral conditions coincide with the equilibrium points of system (A.5), i.e., the points x̄ ∈ X such that x̄ ∈ D(A), H(x̄) ∈ j (X ) and A x̄ + j −1 (H(x̄)) = 0. Theorem A3. ([11, Proposition VII.5.6]) Let x̄ be a steady state of (A.6). Assume H is Fréchet differentiable in x̄ and define B := H (x̄) ∈ B(X, Y ). Then the semiflow v(·; ·) given by (A.6) is uniformly Fréchet differentiable at x̄, i.e. for all t1 > 0 and all ε > 0 there exists δ > 0 such that if ||x − x̄|| < δ and t ∈ [0, t1 ] then ||v(t; x) − x̄ − Dx v(t; x̄)(x − x̄)|| < ε||x − x̄||. The family of linear operators S defined as S(t) := Dx v(t; x̄) : X −→ X for all t ≥ 0, is given implicitly by  t  S(t)x = T (t)x + j −1 T ∗ (t − s)B S(t)xds ∀t ≥ 0, (A.7) 0 and explicitly by the series ∞  S= Sn (A.8) n=0 with S0 = T and  t  Sn (t) = j −1 T ∗ (t − s)B Sn−1 (s)ds ∀t ≥ 0. 0 Moreover, the partial sums in (A.8) converge uniformly on compact intervals. Notice that S(t) is well defined because, since T is closed by ∗-integration on Y and B is continuous and takes values in Y , Theorem A1 ensures the existence of a unique strongly continuous semigroup S which is solution of (A.7). The generator of S is also given by Theorem A1. From the previous theorem it follows that, for all fixed t > 0, v(t; x) can be approximated by x̄ + S(t)(x − x̄) for those x close enough to x̄. It seems a good strategy to infer the behavior of v(·; x) close to x̄ by means of the stability properties of S. However, we must justify carefully the validity of this procedure because, a priori, the asymptotic behavior of S could not determine the stability of the equilibrium x̄. 58 Page 30 of 33 C. Barril and À. Calsina J. Evol. Equ. To our knowledge, it is yet an open question if it could exist nonlinear semiflows with an asymptotically stable equilibrium for which the linearised system around it is unstable. Fortunately, some results in the literature can be applied to our system in order to discard this pathological possibility. Before exposing them let us recall a couple of concepts. Definition A4. The growth bound of a strongly continuous linear semigroup T on a Banach space X is defined as ω0 (T ) := inf{ω ∈ R | ∃M ≥ 1 such that T (t) ≤ Meωt ∀t ≥ 0}. Definition A5. A strongly continuous linear semigroup T is said to be exponentially stable if its growth bound is strictly negative. Similarly, T is said to be exponentially unstable if its growth bound is strictly positive. If S is exponentially stable (i.e. there exists M ≥ 1 and ω > 0 such that S(t) ≤ Me−ωt ), it can be shown that x̄ is a locally asymptotically stable equilibrium (the continuity of v(·; x) makes possible to apply essentially the same argument used in the theory of ODEs [19]). Theorem A4 ([7, Theorem 4.2]). Let x̄, v and S as in Theorem A3. If ω0 (S) < 0 then x̄ is locally asymptotically stable in the Lyapunov sense. More precisely, there exist ω > 0 and δ > 0 such that if x − x̄ < δ, eωt (v(t; x) − x̄) → 0 as t → ∞. In order to give an instability result some additional hypotheses other than ω0 (S) > 0 have to be assumed. Theorem A5 ([7, Theorem 4.3]). Let x̄, v and S as in Theorem A3, and denote A S the generator of S. Assume ω0 (S) > 0 and that X admits a decomposition X = X1 ⊕ X2 (A.9) into S-invariant subspaces with X 1 finite-dimensional. For i ∈ {1, 2} let Si be the restriction of S to X i and let A Si be the corresponding generators. If ω0 (S2 ) < min{Reλ | λ ∈ Spectrum(A S1 )} then x̄ is unstable, i.e. there exist M > 0 and a sequence {xn , tn }n≥1 ⊂ X × R satisfying xn → x̄ and tn → ∞ such that (v(tn , xn ) − x̄ ≥ M. The above results are useful provided we have a method to study the dynamics of S. In general this cannot be done in a straightforward manner because S is given either implicitly or as a series. To overcome this problem the generator A S can be used, since its expression is usually simpler than S and some relations are known between the growth bound of a semigroup T and the spectral bound of its infinitesimal J. Evol. Equ. Semilinear formulation of a hyperbolic system Page 31 of 33 58 generator A T , denoted by s(A T ). In general one can only say that s(A T ) ≤ ω0 (T ), since counterexamples exist in which the strict inequality holds (see [12, Chapter 5] and [26] for a review on asymptotics of semigroups). However, if T is eventually norm continuous, the relation s(A T ) = ω0 (T ) as well as a mapping linking the spectrum of T (t) with the spectrum of A T does hold. Theorem A6 ([12, Theorem IV.3.10]). Let T be an eventually norm continuous semi- group on a Banach space X with generator A T . Then T satisfies the Spectral Mapping Theorem: for all t ≥ 0, the spectrum of T (t) and the spectrum of A T , denoted by σ (T (t)) and σ (A T ) respectively, satisfy the following set relation σ (T (t)) \ {0} = etσ (AT ) . In particular the equality ω0 (T ) = s(A T ) holds. Taking this into account, Theorems A4 and A5 can be modified in terms that they only involve information about the spectrum of A S . Theorem A7. Let x̄, v and S as in Theorem A3, and denote A S the generator of S. If S is eventually norm continuous then (i) x̄ is locally asymptotically stable if s(A S ) < 0, (ii) x̄ is unstable if there exists ω > 0 such that the spectrum of A S within the region {λ ∈ C | Re(λ) > ω} is non-empty and is composed only by a finite number of eigenvalues with finite algebraic multiplicity. Proof. Statement (i) follows directly from Theorem A4 and the relation s(A S ) = ω0 (S) due to the eventually norm continuity of S. To prove statement (ii) we use the decomposition theorem [16, Theorem 6.17] by taking a Jordan curve enclosing the eigenvalues to the right of ω. For full details see [3, Theorem 1.2.21].  As in the standard theory of perturbation semigroups, the property of being even- tually norm continuous is preserved under compact perturbations. Specifically: Theorem A8. Let T be an eventually norm continuous semigroup generated by the linear operator A and closed by ∗-integration on Y ⊂ X T ∗ . Let K be a compact operator from X into Y . 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