MONOTONE SYSTEMS INVOLVING VARIABLE-ORDER NONLOCAL OPERATORS

: In this paper, we study the existence and uniqueness of bounded viscosity solutions for parabolic Hamilton–Jacobi monotone systems in which the diﬀusion term is driven by variable-order nonlocal operators whose kernels depend on the space-time variable. We prove the existence of solutions via Perron’s method, and considering Hamiltonians with linear and superlinear nonlinearities related to their gradient growth we state a comparison principle for bounded sub and supersolutions. Moreover, we present steady-state large time behavior with an exponential rate of convergence.


Introduction
In this paper we are interested in studying the existence and uniqueness of systems with the form (1.1) ∂ t u i +H i (x, t, u(x, t), Du i (x, t), D 2 u i (x, t), T i,x,t u i (x, t)) = 0 in Q, i ∈ I(n), where Q := R d × (0, +∞) and I(n) := {1, 2, . . . , n}. We have considered here the usual notation of bold letters for vectors.
We complement system (1.1) with the initial condition (1.2) u = u 0 := (u 0,i ) i∈I(n) of continuous functions, that is, u 0,i ∈ C(R d ) for all i ∈ I(n). Moreover, for each i ∈ I(n), H i ∈ C(R d × R + × R n × R d × S d × R), and the term T i,x,t (which for ease of notation we call only T i ) is a nonlocal operator playing the role of the diffusion defined as follows: for x ∈ R d , t ∈ R + , and φ : R d → R, a bounded continuous function which is C 2 in a neighborhood of x, we write where B is the open ball centered at zero with radius 1. For each i ∈ I(n), the kernel K i : R d × R + × (R d \ {0}) → R is a continuous function, satisfying and α i : R d × R + × R d → (0, 1) is a continuous function such that α i (x, t, z) =: α + < 1.

(S)
Since for each i ∈ I(n) the operators K i (·) and α i (·) are functions depending on the (x, t, z)-variable, following the ideas of [12], [46], the terms T i are called variable-order nonlocal operators. It is interesting to note that under the symmetric assumption on K i we can remove the compensator term of (1.3). Hence, considering K i (x, t, z) = C i |z| −(d+2αi(x,t,z)) with α i verifying α i (x, t, z) = α i (x, t, −z) for all (x, t, z) ∈ R d × R + × R d , a particular class of operators with the form (1.3) are the variable-order fractional Laplacians given by the expression Moreover, when α i ∈ (0, 1) is a constant, T i reduce to the usual fractional Laplacian operator.
As general conditions we assume that the Hamiltonians satisfy the following properties: (H1) Degenerate ellipticity: for all i ∈ I(n) and (x, t, r, p) ∈ R d × R + × R n × R d H i (x, t, r, p, X, l 1 ) ≤ H i (x, t, r, p, Y, l 2 ) if l 2 ≤ l 1 , Y ≤ X.
(H2) Monotone property: there exists c ij ∈ R for all i, j ∈ I(n) such that n j=1 c ij ≥ c 0 with c 0 ≥ 0 for all i ∈ I(n) and for any (x, t, r, p, X, l) ∈ R d × R + × R n × R d × S d × R and δ > 0, we have c ij δ ≤ H i (x, t, r + δe j , p, X, l) − H i (x, t, r, p, X, l) ≤ 0, if j = i, c ii δ ≤ H i (x, t, r + δe i , p, X, l) − H i (x, t, r, p, X, l), where (e i ) i∈I(n) is the canonical basis of R d .
Note that in (H2) necessarily c ij ≤ 0 for i = j and c ii ≥ 0. A particular case of monotone systems verifying (H2) are the weakly coupled systems with the form for all i ∈ I(n), where c ii (x, t) ≥ 0, c ij (x, t) ≤ 0 for j = i and n j=1 c ij (x, t) ≥ 0.
Concerning the classical local problems of first or second order, the basic assumptions (H1) and (H2) in the nonlocal-term independent formulation make it possible to deal with system (1.1)- (1.2) in the viscosity solution framework; see [23] and [25]. It is well known that this type of systems is compatible with comparison principles, those that together with some regularity properties lead to results related to homogenization, ergodicity, and large time behavior of first and second-order monotone and weakly coupled systems of Hamilton-Jacobi equations; see [18], [34], [38], [40], and references therein.
The notion of viscosity solutions was first introduced by Crandall and Lions [22] and their theory was applied initially to local partial differential equations, but motivated by applications to finance, physical sciences, and mechanics (see for example [15] and [52]), the theory was almost immediately extended to the context of partial integrodifferential equations, i.e., equations involving nonlocal operators such as Lévy operators and their best-known representative, the fractional Laplacian. There is a close connection with probability since nonlocal operators of Lévy type arise as the infinitesimal generator of stochastic Lévy processes and also appear in the context of optimal control of jump diffusion processes; see [16], [42], and [44].
The first paper devoted to the mentioned extension, in the context of stochastic control of jump diffusion processes, was developed by Soner [47]. Following this work, Sayah [45], in the stationary case and using first-order equation techniques, studied stability, comparison results, and the existence of viscosity solutions of a quite general class of integrodifferential equations, nonlinear with respect to the nonlocal term. In the case of bounded measures, Alvarez and Tourin ( [1]) obtained quite general results for parabolic equations. For singular measures several results were obtained, for instance in [5] and [43]. Jakobsen and Karlsen ( [29], [30]) developed a general theory for second-order parabolic nonlinear integro-differential equations, including comparison results, continuous dependence estimates, and a maximum principle. For more advanced theory, concerning regularity, homogenization, and large time behavior of Hamilton-Jacobi problems, we refer to [2], [7], and [48]. It should be noted that the difficulties in the study of the above problems involving Lévy-type operators are, for example, the coupling of second-order derivatives and nonlocal terms, the singularity of the measure appearing in the nonlocal operator, and the lack of basic differentiation tools, such as the product or chain rule. These difficulties were extensively analyzed by Barles and Imbert in [6].
Concerning nonlocal operators of variable order which verify similar conditions to (K)-(S), several properties have been intensively investigated both from the probabilistic and the analytic point of view. Stablelike processes were introduced originally by Bass [9], [10], by showing the uniqueness of solutions to the martingale problem as variants of α-stable processes (said papers considered a spatially dependent index α(x)). Such a process may be thought of as one that at the point x behaves like the symmetric stable process of index α(x), but the index α(x) varies from point to point. Tsuchiya ([49]) defined the processes in terms of stochastic differential equations with jumps. In [28] Jacob and Leopold constructed a Feller semigroup generated by operators with variableorder symbols. Their approach was mainly based on the method of Jacob concerning the generation of Feller semigroups by pseudo-differential operators in [26], [27], and the results of Leopold on pseudo-differential operators of variable order and corresponding function spaces [35], [36], and [37]. See also [31] and [41] for further results concerning the existence of transition densities and path behavior of stable-like processes.
On the other hand, there are numerous results related to problems in PDE's that involve nonlocal operators like (1.3). To mention just one example, using probabilistic methods, in [14] Bass and Levin proved Harnack inequality of harmonic functions with respect to a class of pure jump Markov processes in R d , whose kernels are comparable to those of symmetric stable processes. Bass and Kassmann generalized this result and obtained Hölder continuity of harmonic functions of variable order in [12] and [13]. Bass also established in [11] the Schauder estimates for stable-like operators in R d . With a purely analytical proof, in [46], Silvestre provided regularity results for solutions to integrodifferential equations, including the case of an operator with variable orders. In [17], Caffarelli and Silvestre generalized this result to fully nonlinear integro-differential equations associated with symmetric kernels comparable to fractional Laplacian, and in [32] and [33], Kim and Lee extended this result to equations associated with nonsymmetric kernels. In [3], Bae proved regularity results for solutions of fully nonlinear integro-differential equations with variable-order operators. Also, in [4], Bae and Kassmann established Schauder estimates for problems involving variable-order operators. For other related works, we refer the reader to [20], [21], [39], [50], and references therein. Especially, we mention these new works [51], [53], where the order of the kernel in the variableorder fractional Laplacian verifies (K)-(S) and depends continuously on the variables involved.
The aim of this work is to study the existence and uniqueness of bounded viscosity solutions for the parabolic Hamilton-Jacobi monotone system (1.1)-(1.2) in the framework of viscosity solutions, in which the integro-differential operators are given by (1.3). Moreover, we state a comparison principle for bounded sub and supersolutions and we give steady-state large time behavior with an exponential rate of convergence.
Let us briefly outline the content of this paper. Concerning system (1.1), the novelty is the presence of nonlinearities H i having an explicit interaction with the time variable. Also, the diffusion terms are driven by variable-order nonlocal operators whose kernel depends continuously on the space-time variable. As is usual in the viscosity theory, the existence of discontinuous solutions to (1.1) is performed by Perron's method, adapted in this case to systems involving nonlocal operators with (x, t)-dependent kernels satisfying (K)-(S).
Since we are working with monotone systems and all the kernels K i verify the integrability condition (K1), which allows us to deal with the kernel dependency on the (x, t)-variables (see Section 4), following the standard procedure of doubling variables it is possible to provide a comparison principle for system (1.1). It is important to note that the continuity of the kernels K i and the orders α i are crucial when proving the existence of solutions and the comparison principle. In addition, (K)-(S) allows the integro-differential operators T i to verify the Lévy integrability condition (see [6]), allowing in certain cases the integrability of the kernel. Moreover, to prove the comparison result we concentrate on two main classes of Hamiltonians which are standard in the analysis of fully nonlinear equations, and have been addressed in different contexts to get well-posedness; see for example [6]. The first type of operators are Hamilton-Jacobi functionals with model form where Θ is compact, a θ , A θ , b θ , f θ are bounded and continuous functions, a θ ≥ 0, A θ is positive (semi)definite, and (c ij ) ij satisfies the above conditions. This type of equation fits into assumption (F2); see Section 4.
The second type of problem we are interested in is related to models which are coercive in the gradient with the model form for some m > 1 and A, b, f are continuous and bounded, with b uniformly positive, A is positive (semi)definite, and taking (c ij ) ij as in the previous case. This type of nonlinearity arises in the context of Hamilton-Jacobi equations with unbounded controls and shall be considered as a particular example falling into assumption (F2)'.
In the last section of this article we construct classical sub and supersolutions verifying the initial condition (1.2). Using the comparison principle we get the existence of continuous viscosity solutions. As an application, we use the half-relaxed limits method introduced by Barles and Perthame [8] to conclude steady-state large time behavior for the solution of (1.1)-(1.2) when c 0 > 0 in (H2). Moreover, we find an exponential rate of convergence with a precise index given by c 0 .

Basic notation and viscosity evaluation
In this brief chapter we introduce some notation associated with the viscosity formulation of our system. We represent by B δ (x) the open ball centered at x ∈ R d and radius δ > 0. If x = 0, we write B δ and B if in addition δ = 1. Moreover, the open cylinder in the space-time is defined by For a nonempty set A ⊂ R d and x, p ∈ R d and φ : R d → R, we also consider and we omit the p-term in the case p = Dφ(x), i.e., To end this chapter we give a definition of viscosity solution to system (1.1)-(1.2), which is basically the one given in [23] and [48]. We state the definition in the finite horizon setting, for which we introduce the notation Q T = R d × (0, T ] for T > 0.
, then the following inequality holds: where the last nonlocal dependence is understood as Dφ(x, t)). Finally, u is a viscosity solution of (1.1)-(1.2), if u is a sub and supersolution of the system. Following standard viscosity arguments the above definition can be equivalently stated by assuming that u(x 0 , t 0 ) = φ(x 0 , t 0 ) and that (x 0 , t 0 ) is a strict maximum point. Moreover, in order to avoid technicalities in the viscosity evaluation, the following lemmas will be useful in the sequel.
Proof: We focus on the "maximum" statement, hence Taking σ ≤ δ and since the compensator term plays no role because it is common to both sides of the inequality, we see that Lemma 2.3. Definition 2.1 can be equivalently formulated by taking i ∈ I(n), the test function φ ∈ C 2 (Q T ) bounded, and (x 0 , t 0 ) a global maximum (resp. minimum) point of u i − φ inQ T and instead of the viscosity inequality we write Proof: By (H1), the general setting implies the new version. To prove the converse we focus only on the case of the subsolution. We consider Applying the dominated convergence theorem, we can pass to the limit and get . By the continuity of H i , we conclude the desired inequality.

Existence of discontinuous solutions
Before stating our first results, we need to introduce extra notations. We write u * for the upper semicontinuous envelope of u, which is performed for each component, and similarly u * for the lower semicontinuous envelope. Also, we give the following definition of discontinuous viscosity solution.
Definition 3.1. A bounded function u :Q T → R n is a discontinuous viscosity subsolution (supersolution) of (1.1) if u * is a subsolution (u * is a supersolution) of (1.1) in the sense of Definition 2.1. We say that u is a discontinuous viscosity solution to (1.1) if it is simultaneously a discontinuous sub and supersolution.
In what follows we perform Perron's method for nonlocal operators developed by Ishii in [24], adapted to systems where the diffusion terms are driven by variable-order nonlocal operators whose kernels depend on the space-time variable. Throughout this section we make assumptions (H1), (H2), and (K)-(S).
Theorem 3.2. Assume that U ∈ LSC(Q T ) and V ∈ USC(Q T ) are bounded discontinuous viscosity sub and supersolutions to problem (1.1) respectively. Then, there is a discontinuous viscosity solution u : Proof: We consider functions z :Q T → R n verifying the condition and we set Λ = {z :Q T → R n | z * is a subsolution of (1.1) and verifies (3.1)}.
We start by noticing that Λ = ∅ since U is a discontinuous viscosity subsolution and verifies (3.1). Hence, we define First we prove that u defined as above belongs to Λ. Indeed, it is direct to see that u verifies (3.1), and to prove that u is a viscosity subsolution . By definition of u * i and Lemma 2.10 of [19], there are sequences . By the previous arguments and since v k is a subsolution of (1.1), we have Now we focus on the nonlocal operator. The continuity of K i and (K)-(S) imply and by the smoothness of ψ it is possible to apply the dominated convergence theorem to conclude that Also, using the definition of u we have that u and since the last term is integrable in B c δ , we have that I 1 → 0 as k → +∞.
For the second term, since u * i (y k +z, t k )−u * i (x 0 , t 0 ) ≥ −2|u| ∞ , by (3.5) and applying Fatou's lemma in its dominated version, we have that Finally, since Dφ and K i are continuous, (y k , t k ) → (x 0 , t 0 ), and by the dominated convergence theorem we have that Joining the previous calculations, we get that ). Thus, by (3.4) and (3.6) we conclude that , (H2), and the definition of u, we notice that .7) and (H1) we have ) ≤ 0, and we conclude that u is a viscosity subsolution of (1.1).
In what follows we prove that the u * given by (3.2) is a viscosity supersolution of (1.1). By contradiction, we assume the existence of i ∈ for some θ > 0.
Since V is a supersolution to (1.1), (3.8) implies that (u i ) * (x 0 , t 0 ) < V i (x 0 , t 0 ). Then, by considering r, > 0 smaller if necessary we have Moreover, by the smoothness and boundedness of φ and since the kernel K i is continuous and satisfies (K)-(S), the function (x, t) → T i φ(x, t) is continuous. Using this, the continuity of H i , (H2), the semicontinuity of u * , and in view of (3.8), for r > 0 small in terms of θ we have We consider the function w defined as w j = u j for j = i and By the above considerations, since U ≤ u ≤ V inQ T and V i > φ + in C r (x 0 , t 0 ), then w verifies (3.1). In what follows, we prove that w * is a viscosity subsolution to (1.1) and such that w i > u i at some point inQ T close to (x 0 , t 0 ). This contradicts the maximality of u and concludes the proof. For this, we consider j ∈ I(n), (x,t) ∈ Q T , ϕ ∈ C 2 (Q T ), and σ > 0 such that (x,t) is a strict maximum point of w If j = i, by definition of w we have that w j = u j , and since u * is a subsolution, we have that , hence, by (H1) and (H2), we have that , similarly to the previous computations, we get the subsolution's viscosity inequality for w * at (x,t). When It is straightforward to see that and concerning the nonlocal terms it is direct to get from the definition ofφ that Now, by (3.5), we notice that Replacing this into (3.10) and using the continuity of H i , taking small in terms of σ and θ we get thatφ satisfies the inequality  Now, since w * i (x,t) > u * i (x,t), we necessarily have that w * i (x,t) = φ(x,t)+ , and taking 0 < σ < min(σ , σ) small enough we have that and the smoothness of φ and ϕ imply that From this we can get Adding the above inequalities, using that w * i ≥ φ and the definition ofφ, we conclude that T i,σ (w * i , ϕ,x,t) ≥ T iφ (x,t). Then, using the last inequality, (3.11), (3.12), the properties (H1), (H2), and the definition of w, we arrive at ∂ t ϕ(x,t) + H i (x,t, w * (x,t), Dϕ(x,t), D 2 ϕ(x,t), T i,σ (w * i , ϕ,x,t)) < 0. Finally, applying Lemma 2.2 and (H1), we obtain the viscosity inequality associated with σ.

Comparison principle
In order to state the comparison principle we present the main assumptions on the Hamiltonian H and the kernel K, which are standard in the analysis of fully nonlinear equations.
In the first place, we state a regularity assumption of the Hamiltonian.
(F1) H(x, t, r, p, X, l) is Lipschitz continuous in the nonlocal variable l, uniformly with respect to the other variables (x, t, r, p, X).
Also, we divide the proof of comparison into two main cases, mostly related with the behavior of the nonlinearity H in terms of the gradient.
The first type is a linear gradient growth satisfying the following condition.
(F2) For all R > 0 and T > 0, there exist moduli of continuity ω, ω R,T such that for all |x|, |y| ≤ R, |r| ≤ R, s, t ∈ [0, T ], l ∈ R, and X, Y satisfying the matrix inequality The second type is superlinear Hamiltonians. (F2)' There exist m > 1 and C > 0 such that, for all R > 0, T > 0, and all µ ∈ (0, 1), there exist moduli of continuity ω, ω R,T such that for all |x|, |y| ≤ R, |r| ≤ R, t ∈ [0, T ], l ∈ R, and X, Y satisfying We can see, as a simple example, that the kernel of the fractional Laplacian K(x, t, z) = C|z| −d−2α with constant α ∈ (0, 1) verifies trivially (K1). For more developed examples of functions that verify these conditions we can consider the following two cases: In the first one, if K verifies (K)-(S) with K(x, t, z) ≡ C|z| −d−2α(x,t,z) , satisfying the assumption (K1) is enough to consider K(x, t, z) locally Lipschitz in the variable (x, t), uniformly with respect to the variable z.
On the other hand, if K(x, t, z) = C|z| −d−2α(x,t,z) , to satisfy assumption (K1), we can consider for example At this moment, it is useful to consider the following technical results. Hence, we introduce some notation and we define , is a viscosity subsolution to the system for some constant C > 0 depending on T , |u| ∞ , and (c ij ) i,j∈I(n) of (H2).
Proof: Let i ∈ I(n), δ > 0, (x 0 , t 0 ) ∈ Q T , and φ ∈ C 2 (Q T ) such that w i − φ attains a maximum in (x 0 , t 0 ). We split our analysis into two cases: If i ∈ I 1 and since u is a subsolution of (1.1), we have In the following computations, we drop the dependence of H i on (x 0 , t 0 , Dφ, D 2 φ, T i,δ (w i , φ, x 0 , t 0 )), since these variables do not play any role here. Also, we denote by (e i ) i∈I(n) the canonical basis of R d .
Hence, taking j ∈ I 2 , since j = i by (H2), there exists c ij ≤ 0 such that By (4.3) and taking C j = −c ij |u| ∞ > 0 we notice that Now, if j ∈ I 1 (including the case j = i), we have that w j = u j −ηt ≤ u j , and by (H2) By the above computations and iterating the procedure for each j ∈ I(n), we conclude the desired viscosity formulation since for a constant C > 0 large enough and c 0 ≥ 0, the constant that appears in (H2). Now, if i ∈ I 2 , using the fact that u is a subsolution of (1.1), we have Doing a similar analysis as in the previous case, for all j ∈ I 1 by (H2) we note that By (4.4), there exists a constant C j > 0 such that Now, if j ∈ I 2 (including the case j = i), we have that µ −1 w j = u j − µ −1 ηt ≤ u j , and by (H2) Thus, by (4.4) there exists C j > 0 such that Iterating the procedure for each j ∈ I(n), we conclude for a constant C > 0 large enough.
Before stating the comparison principle and following the spirit of [6], we introduce some properties of localization terms in the state variable.
Then, ψ β satisfies Proof: We concentrate on the nonlocal operator since the other results are direct. For each (x, t) ∈ R d × R + , by (K)-(S) we have At this moment we are able to establish the comparison principle for bounded viscosity sub and supersolutions.
Proof: We assume by contradiction that By taking β, η > 0 small enough and µ close to 1 in terms of T , Θ, |u| ∞ , |v| ∞ , using the notation of Lemma 4.1 we consider w defined as in (4.2), hence we get At this point we start the standard process of doubling variables. For 0 < < γ we define for all (x, t), (y, s) ∈Q T and i ∈ I(n). Then, by (4.5) and the definition of Ψ there exists a point (x,ȳ,s,t,ī) such that Since I(n) is finite, up to a subsequence if necessary, we can assume the indexī above is fixed and independent of the other parameters, namely i. Then, using the boundedness, the semicontinuity of w, v, and the maximal condition above we see that Moreover, the localization term ψ β makes |x|, |ȳ| ≤ 4/β for all γ, small enough. Then, up to a subsequence there exists (x * , t * ) ∈Q T such thatt,s → t * andx,ȳ → x * as , γ → 0. And using that u ≤ v at t = 0 it is possible to conclude thats,t > 0 uniformly in , γ when β > 0 is fixed small and µ < 1 is fixed close to 1 in terms ofΘ. Now, in order to establish the viscosity evaluations, we denote which serves as a test function for w at (x,s), and which serves as a test function for v i at (ȳ,t). Thus, applying the Crandall-Ishii-Lions lemma for nonlocal problems given by Corollary 1 in [6], properly adapted to our current framework, and by Lemma 4.1 we can write , and X, Y are symmetric matrices satisfying condition (4.1), for any δ > 0 small enough. Taking into account that ∂ t φ 1 (x,s) = ∂ t φ 2 (ȳ,t) and subtracting the above equations we obtain that . Calling the first term of the above inequality A, we have Now, by Lemma 4.2 we have thatq =p+o β (1) with o β (1) → 0 as β → 0. Hence, to estimate B 1 we divide the analysis into cases: if H i satisfies (F2), then µ i = 1, hence by applying (F2) directly we see that If H i satisfies (F2)', then µ * = µ and we see that Then, ifp is uniformly bounded when γ is small, it is easy to see that Otherwise, ifp is unbounded, for µ < 1 fixed, we consider γ and for all γ, β small enough in terms of 1 − µ, such that (4.6) implies Then, summarizing the above estimates and considering , γ, β as above, by (4.6) we conclude Now, before dealing with the term A 2 , we analyze the nonlocal operators. Hence, by definition of φ 2 and (K)-(S), we have Similarly, Moreover, by the maximality of (x,ȳ,s,t) we see that where I(w i ,x,s,ȳ,t) Now we concentrate to get a bound for the term I, hence For the terms I 1 and I 2 , by (K1) we notice that and Hence, by the dominated convergence theorem we have that By the above computations and (4.6), we obtain that T i (w i ,x,s,ȳ,t) = o γ (1) as ε γ and γ → 0. Hence, if we join the approach of I with (4.7), (4.8), and (4.9), by Lemma 4.2 we get that Now, by (4.10), (H1), and (F1), we have To deal with the last term we use the monotone property (H2). Then, we define φ 1 ,x,s)). Hence, By definition of Ψ, w, and v, we have that therefore, using (H2) for all k = i we obtain and since c ik ≤ 0 for all k = i, by (4.11) we have Again, by (H2) and (4.11) we have

Then, we have
Finally, replacing the bounds of A in (4.7), we obtain Fixing η > 0 and letting δ, , γ, β → 0 and µ → 1, we arrive at a contradiction. This concludes the proof.

Well-posedness and large time behavior
The main result of this paper is the following existence and uniqueness for system (1.1)-(1.2). Moreover, if c 0 > 0 in (H2), the solution u is uniformly bounded and we have the estimate for some constant C > 0 depending on c 0 .

Proof:
We begin by considering Q T with T > 0. First we assume that u 0,i ∈ C 2 b (R d ) for all i ∈ I(n), with |u 0 | C 2 (R d ,R n ) < +∞. Then, we consider V i (x, t) = Ct + u 0,i (x) for some C > 0 to be fixed later, for all i ∈ I(n).
By (K)-(S) we see that for some R > 0 fixed. By (F3) and for each i ∈ I(n), there exists a constant C H,i (R) > 0 such that for all (x, t) ∈ Q T . Taking C > 0 large in terms of max i∈I(n) {C H,i (R)}, we thus get that V is a classical supersolution to (1.1) for the problem set up on Q T , which satisfies V(x, 0) = u 0 (x) for all x ∈ R d . Similarly, the function U i (x, t) = −Ct + u 0,i (x) for all i ∈ I(n) is a classical subsolution to (1.1) for some C > 0 large enough verifying U(x, 0) = u 0 (x). Hence, Theorem 3.2 implies the existence of a discontinuous viscosity solution u for (1.1) such that u * (x, 0) = u * (x, 0) = u 0 (x) for all x ∈ R d and thus by Theorem 4.3 we conclude u ∈ C(Q T ). Uniqueness follows again by Theorem 4.3.
In the general case, when Thus, by the previous analysis we can construct a supersolution V i (x, t) = u 0,i (x) + + C t, and a subsolution U i (x, t) = u 0,i (x) − − C t to equation (1.1). Then, defining by the first part of the proof of Theorem 3.2 we have that U, V are respectively viscosity sub and supersolutions to problem (1.1) and match with u 0 at t = 0. From this point we follow the same lines of the previous case.
Finally, given 0 < T < T , the viscosity solution in (0, T ] must coincide in (0, T ] with the viscosity solution in this interval, by uniqueness. Thus, the viscosity solution of (1.1)-(1.2) extends uniquely to all t ∈ [0, +∞), i.e., it is global in time.
For the estimate (5.1), we consider the constant function V i = C + |u 0 | ∞ > 0 for all i ∈ I(n), hence by (H2) for each j ∈ I(n) we have Furthermore, iterating the process we obtain and taking C = c −1 0 |H(·, ·, 0, 0, 0, 0)| ∞ , we get that V is a supersolution to (1.1)-(1.2) for the problem on Q. A lower bound can be obtained similarly by considering a function with the form U i (x, t) = −C − |u 0 | ∞ for all i ∈ I(n). By the comparison principle, the proof is complete.
The comparison principle is used to study steady-state large time behavior for system (1.1). More specifically, we consider functionsH i andK i (with the associated functionᾱ i ) not depending on the time variable t and satisfying the above conditions in the time-independent formulation for all i ∈ I(n). Thus, the stationary problem has the form Following closely the lines in the proofs of Theorem 3.2 and Theorem 4.3, where c 0 > 0 in (H2) plays the role of the parabolicity performed by the time derivative, we can conclude the well-posedness of (5.2), given the following result. |u| ∞ ≤ c −1 0 |H(·, 0, 0, 0, 0)| ∞ . The uniqueness for the stationary problem leads to a steady-state large time behavior result for parabolic problems as an application of the half-relaxed limits method introduced by Barles Proof: Let u be the solution to (1.1). For each i ∈ I(n), u i is a bounded function in Q from which the functions are well defined for each (x, t) ∈ Q. It is worth noting that lim sup and analogously for lim inf.
We claim that for each t 0 > 0, the functions x →ū(x, t 0 ) and x → u(x, t 0 ) are respectively viscosity sub and supersolutions to (5.2). We will only deal with subsolutions, as the case of supersolutions is similar.
Let t 0 > 0 fixed and i ∈ I(n), x 0 ∈ R d , φ ∈ C 2 b (R d ) such that u i (·, t 0 ) − φ has a global, strict maximum point at x 0 . Hence, taking appropriate sequences in the formulation ofū i , we have the existence of δ > 0, x k → x 0 , and k → 0 such that, denoting t k = t 0 / k , (x k , t k ) is a maximum point of the function (x, t) → u i (x, t) − φ(x) in B δ (x 0 ) × (t k − δ/ k , t k + δ/ k ), satisfying in addition that u i (x k , t k ) →ū i (x 0 , t 0 ) as k → ∞. Now, since u is a subsolution to (1.1) and since ∂ t φ(x) = 0, we can write (5.4) H i (x k , t k , u(x k , t k ), Dφ(x k ), D 2 φ(x k ), T i,δ (u i , φ, x k , t k )) ≤ 0.
Before continuing we analyze the nonlocal term; hence by (K)-(S), (5.3), and the dominated convergence theorem we have whereT i is the operator (1.3) with the kernelK i . Also, we notice that Similarly to the previous case, by using the dominated convergence theorem, the properties (K)-(S), and the uniform convergence of (5.3), we have I 1 , I 3 = o k (1). For the term I 2 we conclude I 2 ≤ o k (1) by using the definition ofū i and (K)-(S). Then, summarizing the above estimates, Using the property (H1) and (5.4), we have Then, considering the above estimate, the smoothness of φ, the continuity of H i , taking k → +∞, since then u i (x k , t k ) →ū i (x 0 , t 0 ), (5.3), the definition of eachū j , and the monotone property (H2), we conclude that which is the desired viscosity inequality.
To conclude the steady-state large time behavior, we apply the comparison principle for the stationary problem (5.2) and we arrive atū ≤ u, which leads to the local uniform convergence of u to the unique solution of the stationary problem. Now, in order to prove the rate of convergence, we notice that, since H, K do not depend on t, we have that H =H, K =K in (5.3). Hence, we consider the function U i (x, t) = u ∞,i (x) + e −c0t |u ∞ − u 0 | ∞ for all i ∈ I(n) and c 0 > 0 is the constant in (H2).
In the same way, the function is a subsolution to (1.1), and the result follows by the comparison principle.