ANISOTROPIC PARABOLIC EQUATIONS WITH VARIABLE NONLINEARITY

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p ( x,t )-Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L ∞ bounds for the weak solutions.

1. Introduction 1.1. Statement of the problem and assumptions. Let Ω ⊂ R n be a bounded simple-connected domain and 0 < T < ∞. We consider the Dirichlet problem for the parabolic equation where z = (x, t) ∈ Q T ≡ Ω × (0, T ], Γ T is the lateral boundary of the cylinder Q T , D i denotes the partial derivative with respect to x i and The coefficients a i (z, u), b i (z, u) and d(z, u) may depend on z = (x, t), u(z) and obey the following conditions: (1.2) a i (z, r), b i (z, r), d(z, r) are Carathéodory functions (defined for (z, r) ∈ Q T × R, measurable in z for every r ∈ R, continuous in r for a.a. z ∈ Q T ), ∀ (z, r) ∈ Q T × R 0 < a 0 ≤ a i (z, r) ≤ a 1 < ∞, a 0 , a 1 = const, (1.3) with positive constants b, d 0 , d 1 , d 2 , λ > 1, and The exponents p i (z) are given continuous in Q T functions such that , with finite constants p ± , p ± i > 1. Moreover, it will be assumed throughout the paper that the exponents p i (z) are continuous in Q T with logarithmic module of continuity: where lim τ →0 + ω(τ ) ln 1 τ = C < +∞. with the constant exponent of nonlinearity p ∈ (1, ∞). During the last decades equation (1.8) was intensively studied and was casted for the role of a touchstone in the theory of nonlinear PDEs. There is extensive literature devoted to equation (1.8). We limit ourselves by referring here to monographs [24], [36], papers [5], [9], [20], [29] and the review paper [30] which provide an excellent insight to the theory of evolutional p-Laplacian equations. PDEs with variable nonlinearity are very interesting from the purely mathematical point of view. On the other hand, their study is motivated by various applications where such equations appear in the most natural way. Equations of the type (1.1) and their elliptic counterparts appear in the mathematical descriptions of motions of the non-newtonian fluids [11], in particular, electro-rheological fluids which are characterized by their ability to change the mechanical properties under the influence of the exterior electro-magnetic field [27], [39], [40]. Most of the known results concern the stationary models, see, e.g., [1], [2], [3]. Some properties of solutions of the system of modified nonstationary Navier-Stokes equations describing electro-rheological fluids are studied in [4]. Another important application is the image processing where the anisotropy and nonlinearity of the diffusion operator and convection terms are used to underline the borders of the distorted image and to eliminate the noise [6], [8], [21]. Many of the frequently discussed schemes of image restoration lead to nonlinear elliptic and parabolic equations with linear growth in the diffusion operator; this situation corresponds to the case p − = 1 and is not discussed in the present paper.
To the best of our knowledge, the reported results on the solvability of parabolic equations of the type (1.1) concern the equations with linear growth at infinity whose solutions are understood as elements of the space L 2 0, T ; BV (Ω) ∩ L 2 (Ω) , see, e.g., [7], [8], [21]. In our assumptions on the structure of the equation, the weak solutions possess better regularity and belong to Orlicz-Sobolev spaces W 1,p(·) (Q T ) (the rigorous definition is given in Section 2 below). Moreover, it is proved in [18] that the gradient of the solution to the evolutional p(x, t)-Laplacian satisfy the Meyer-type estimate: the gradient is integrable with the exponent p(z)(1 + δ), δ > 0, instead of p(z) as is prompted by the equation. It is known also that the solutions of equation (1.1) may extinct in a finite time [15], [17], a property typical for the solutions of the fast diffusion equation. In contrast to the case of the fast diffusion equation with constant exponents of nonlinearity, the variable nonlinearity makes that this property may persist even if the equation eventually transforms into the linear one. It is worth mentioning here the papers [31], [32], [33] devoted to the study of similar effects in solutions of equations with singularly perturbed coefficients and exponents of nonlinearity.
Parabolic equations with variable nonlinearity of the type are studied in papers [12], [16]. This class of equations generalizes the famous porous media equation (PME) to the case of variable exponents of nonlinearity. It is shown in [12] that the weak solutions of this equation display many of the properties intrinsic to the solutions of PME. However, the methods used in the study of solvability of such equations are specific for the generalized PME and can not be directly applied to equations of the type (1.1) which are nonlinear with respect to D i u.
Stationary counterparts of equation (1.1) and the generalized PME were studied by many authors. We refer here to [13], [14], [37] for a review of the relevant results.
1.3. Organization of the paper and description of results. The paper is organized as follows. In Section 2 we introduce the function spaces of Orlicz-Sobolev type and present a brief description of their main properties. In our conditions on the regularity of the data, the smooth functions are dense in these spaces, which allows us to construct a solution using the sequence of Galerkin's approximations.
The main existence result for problem (1.1) is stated in Theorem 3.1. We prove that problem (1.1) has at least one global weak solution if the growth conditions (1.4) and (1.6) are fulfilled with 2 ≤ λ = max{2, p − −δ} for some δ > 0. The assertion remains true if λ = max{2, p − }, but under the additional condition of smallness of the data u 0 , h d and h b in the corresponding norms. The case λ > max{2, p − } is studied in Theorem 3.2. We show that in this range of exponents, and with the functions h b , h d satisfying (1.5), problem (1.1) has a local in time solution if the parameters λ, p − and n are subject to the conditions max{2, p − } < λ < p − 1 + 2 n , The proofs of these assertions do not require monotonicity of the term d(z, u). The monotonicity of the diffusion part of the equation is used to prove the convergence of Galerkin's approximations. In Section 7 we briefly discuss the possibility of extension of the existence results to the case of homogeneous Neumann boundary condition. Section 4 is devoted to derivation of L ∞ bounds for the solutions of problem (1.1). We assume that the functions h d , h b are subject to the stronger restrictions Under these assumptions we prove in Theorem 4.1 that the weak solutions of problem (1.1) are globally bounded. The growth restriction can be relaxed for the terms d(z, u) of special form. Namely, if we assume that in the foregoing assumptions and that the inequality holds in Q T for some R > 0, then the solutions of (1.1) are globally bounded. Moreover, once such a bound is established, we use it to prove the existence of a global weak solution applying Theorem 3.1. We finally drop conditions (1.9) and show the under assumptions (1.10) problem (1.1) admit a local bounded solution for every λ ≥ 1.
Uniqueness of weak solutions is studied in Section 5. It is shown that the weak solution of problem (1.1) is unique if the function u → d(z, u) is monotone increasing and with the module of continuity ω satisfying the condition ǫ ds ω α (s) → ∞ as ǫ → 0 for some 1 < α < (p + ) ′ = p + p + − 1 .
If the omit the condition of monotonicity of d(z, u), the uniqueness of weak solutions still can be proved but under stronger continuity and growth assumptions: d(z, u) is Lipschitz-continuous with respect to u and ω α (s) = Cs 2 . In the proof of uniqueness we follow ideas of [10], [14], [22], [23] were similar arguments were applied to the study of elliptic equations with nonstandard growth conditions.
In Section 6 we study the dependence of the regularity of solutions to problem (1.1) on the regularity properties of the exponents p i , a i and σ in the partial case when We show that if u 0 ∈ L σ(·,0) (Ω), D i u 0 ∈ L pi(·,0) (Ω), and if the exponents p i and σ are nonincreasing functions of t, then the solutions of problem (1.1) possess better regularity properties: In the concluding Section 7 we give certain extensions of the results to other classes of equations close to (1.1).
2.2. Spaces L p(·,·) (Q T ) and anisotropic spaces W(Q T ). Let p i (z) satisfy conditions (1.6) and (1.7). For every fixed t ∈ [0, T ] we introduce the Banach space holds, which is why V t (Ω) is reflexive and separable as a closed subspace of X. By W(Q T ) we denote the Banach space Let v = (v 1 , . . . , v n ), p(z) = (p 1 (z), . . . , p n (z)), and The following counterpart of (2.3) holds: , |∇u| ∈ L p + (Ω) . Since V + (Ω) is separable, it is a span of a countable set of linearly independent functions {ψ k (x)} ⊂ V + (Ω). Without loss of generality, we may assume that this system forms an orthonormal basis of L 2 (Ω).
Proof: In our conditions on ∂Ω and p i , for every u ∈ V t (Ω) there is a sequence u δ (·, t) ∈ C ∞ (Ω) such that supp u ǫ (·, t) ⋐ Ω and Such a sequence is obtained via convolution of u with the Friedrics's mollifiers [45, It follows now that for all sufficiently large k and small δ Proof: In view of Proposition 2.1, the assertion immediately follows because the functions m k=1 d k (t)ψ k (x) are dense in L p + (0, T ; W 1,p + (Ω)) ∩ L 2 (0, T ; L 2 (Ω)).
Let ρ be the Friedrics mollifying kernel Given a function v ∈ L 1 (Q T ), we extend it to the whole R n+1 by a function with compact support (keeping the same notation for the continued function) and then define as h → 0 by virtue of Proposition 2.3.

Proposition 2.5 (Integration by parts
Take For every k ∈ N and h > 0 The last two integrals on the right-hand side exist because v h , w h ∈ L 2 (Q T ). Letting h → 0, we obtain the equality According to Propositions 2.3 and 2.
In the same way we check that By the Lebesgue differentiation theorem

Existence theorems
In this section we prove the existence of weak solutions to problem (1.1) under the general growth conditions (1.4). The solution of problem (1.1) is understood in the following sense.
and every t 1 , t 2 ∈ [0, T ] the following identity holds: The following are the main results of this section. 3) the constant λ satisfies the condition Then for every u 0 ∈ L 2 (Ω) problem (1.1) has at least one weak solution u ∈ W(Q T ) satisfying the estimate b) The assertion remains true if (3.2) is substituted by the condition λ = max{2, p − } and the constant b 0 + d 0 in (1.4) is appropriately small in comparison with a 0 .
Theorem 3.2. Let us assume that in the conditions of Theorem 3.1 condition (3.2) is substituted by the following one: Then there exists T 0 > 0, defined through u 0 2 L 2 Ω + K, such that problem (1.1) has at least one weak solution u ∈ W(Q T0 ) satisfying estimate The approximate solutions to problem (1.1) are sought in the form where the coefficients c If the coefficients a i , b i , d and the exponents p i , σ satisfy the conditions of Theorem 3.1 a), the functions F k are continuous in all their arguments.

A priory estimates.
Lemma 3.1. Let the conditions of Theorem 3.1 a) be fulfilled. Then for every T < ∞ and m ∈ N system (3.6) has a solution c (m) on the interval (0, T ) and the corresponding function u (m) satisfies the estimate Proof: By Peano's Theorem, for every finite m system (3.6) has a solution c Using (1.3), (1.4) and applying Young's inequality, we estimate: with a constant C depending on ǫ, ǫ d , a 0 , p − , p + . Plugging (3.9)-(3.10) into (3.8), choosing ǫ sufficiently small and simplifying, we get the esti- (3.11) Let λ = 2. Using Gronwall's inequality to estimate the function u (m) (·, t) 2 2,Ω and then reverting to (3.8), we obtain the required estimate (3.7).
Let 2 < λ = p − −δ. This assumption yields the inequality λ < n(p − −δ) n−p − +δ , which allows one to make use of the embedding theorem in Sobolev spaces: Applying now (2.3) and Young's inequality, we arrive at the inequality Gathering these estimates with (3.8) and choosing ǫ appropriately small, we obtain the inequality The right-hand side of the obtained estimate does not depend on m, which is why the solution of system (3.6) can be continued to the maximal interval [0, T ]. Proof: We only have to study the case λ = p − . Then the Poincaré inequality yields Combining (3.11) with this inequality, we have that (3.14) The conclusion follows if we claim that C(b 0 + d 0 ) < a 0 and choose ǫ d sufficiently small. Proof: Instead of (3.12), we will make use of the interpolation inequality The inclusion θ ∈ (0, 1) follows from condition (3.4): Applying (2.3) and Young's inequality we transform (3.15) to the form with the exponents Gathering (3.18) with (3.8)-(3.10) and choosing ǫ appropriately small, we have Introducing the function we write the last inequality in the form The functions satisfying this inequality are bounded on the intervals [0, t 0 ] with (3.20) t 0 e (γ−1)Bt0 < min 1 Since t 0 → ∞ as A → 0, estimate (3.7) takes the form  Proof: Let be a subspace of the set of admissible test-functions. Take a function By the definition of u (m) (see (3.6)) Using (1.4) and (3.7), we may estimate the right-side of this equality as follows: with the constants C ′ and C independent of m. It follows that for The following inclusions hold: It follows that the sequence {u (m) } contains a subsequence strongly convergent in L q (Q T ) with some q > 1 [43]. This subsequence contains a subsequence which converges to u a.e. in Q T (see, e.g., [35, Theorem 2.8.1]). These conclusions together with the uniform in m estimates (3.7) allow one to extract from the sequence u (m) a subsequence (for the sake of simplicity we assume that it merely coincides with the whole of the sequence) such that By the method of construction, each of the functions u (m) satisfies identity (3.1) with the test-function η ∈ Z m . Let us fix an arbitrary m ∈ N. Then for every s ≤ m and η ∈ Z s Letting m → ∞ and using (3.21) we conclude that ∀ η ∈ Z s with an arbitrary s ∈ N. It follows then that identity (3.22) holds for every η ∈ W(Q T ). It remains to identify the limit functions A i . Proof: We rely on the monotonicity of the operator M(s) = |s| p−2 s: According to (3.24) Gathering (3.22) with this inequality, integrating by parts the term u (m) η t , and then letting m → ∞ we conclude, following [36, pp. 158-161], that Choosing now ξ = u ± ǫζ with ǫ > 0, simplifying and then letting ǫ → 0, we have We have shown that under the conditions of Theorems 3.1 and 3.2 the solution u ∈ W(Q T ) satisfies the identity Applying Proposition 2.5 and integrating by parts in the first term of (3.25), we complete the proof of Theorems 3.1 and 3.2.
with finite nonnegative constants d 0 , b 0 . If u 0 ∞,Ω < ∞, then the weak solution of problem (1.1) is bounded and satisfies the estimate Proof: Let us fix k ∈ N and consider the auxiliary problem with d k (z, u) ≡ d(z, min{|u|; k} sign u). Since for every finite k |d (z, min{|u|; k} sign u) | ≤ d 0 k λ−1 + h d , it follows from Theorem 3.1 that problem (4.4) has a weak solution u(z) in the sense of Definition 3.1. Let us introduce the function The function u 2m−1 k with m ∈ N can be taken for the test-function in (3.25). Let in (3.25) Dividing the last equality by h and letting h → 0, we have that ∀ a.e. t ∈ (0, T ) 1 2m (4.5) Indeed: by Lebesgue's dominated convergence theorem for every φ ∈ L 1 (0, T ) and a.e. t ∈ (0, T ) there exists lim h→0 t+h t φ(τ ) dτ = φ(t). Let us write (4.5) in the form: ∀ a.e. t ∈ (0, T ) Integrating by parts, we find that i . and I as follows: .

Let us introduce the function
Choosing ǫ sufficiently small and substituting the above estimates into (4.6), we arrive at the inequality for the function y k (t): Multiplying this inequality by e −C0t , C 0 = (b 0 + d 0 ), and integrating over the interval (0, t) we arrive at the estimate Remark 4.1. It is worth mentioning here paper [44] which addresses the question of local boundedness of solutions to equation (1.1) with anisotropic but constant growth conditions. The method of proof is based on application of suitable embedding theorems in the anisotropic Sobolev spaces.

Global existence via boundedness.
Let us consider the case when in equation (1.1) the term d(z, u) is of the special form: If σ(z) and λ satisfy conditions (1.3), (1.4), the existence of a weak solution follows from Theorems 3.1 and 3.2. If we additionally assume that the conditions of Theorem 4.1 are fulfilled, then this weak solution is bounded. We now turn to the study of the case which does not fall into the scope of Theorems 3.1, 3.2, and 4.1. Let us take a positive number Because of condition σ − > λ > 2, such a number always exists, provided that In this case u ∞,QT ≤ sup Ω |u 0 | + h d ∞,QT .
Proof of Theorem 4.2: Fix an arbitrary finite number R > 0 and consider the regularized problem (4.14) The regularized problem (4.14) has a global weak solution. Moreover, since b i satisfy the conditions of Theorem 4.1, this solution is globally in time bounded: u ∞,QT ≤ C(R). The theorem will be proved if we show that the constant C(R) is in fact independent of R. Let us set with R 0 satisfying the inequality P(z, R 0 ) ≥ 0. Let us take for the test-function in (3.25) the function Arguing like in the proof of Theorem 4.1 we arrive at the equality which can be written in the form ∀ a.e. t ∈ (0, T ) In the last relation The terms J (j) i are estimated exactly like in the proof of Theorem 4.1: Further, Gathering these estimates we find that Since P(z, R 0 ) ≥ 0 by the choice of R 0 , and u + (x, 0) = 0 by the choice of R, the last inequality yields ∀ a.e. z ∈ Q u + (z) = 0, whence u(z) ≤ R a.e. in Q T . The same argument shows that and, finally, This inequality means that which completes the proof.

4.3.
Local existence via boundedness. Let us consider problem (1.1) with the term d(z, u) satisfying the growth condition For 0 ≤ λ ≤ 2 the existence of a global bounded solution to problem (1.1) is proved in Theorem 3.1. The next theorem asserts the existence of local bounded solution in the case λ > 2. such that in the cylinder Q θ problem (1.1) has at least one weak solution u ∈ W(Q θ ) such that u t ∈ W ′ (Q θ ) and u ∞,Q θ < ∞. The solution can be continued to the interval [0, T * ], where Proof: Let us consider the auxiliary problem in Ω with the right-hand side As in the proof of Theorem 4.1, we will make use of the fact that By Theorems 3.1 and 4.1, for every r > 1 the regularized problem (4.18) has a global bounded weak solution u(z). Let us show that the function w(t) = u(·, t) ∞,Ω can be estimated by a constant which does not depend on r. Following the proof of Theorem 4.1 we find that the solution of (4.18) satisfies inequality (4. For every fixed r > 1 R(r, t) → u 0 ∞,Ω as t → 0, whence for every r ≥ u 0 ∞,Ω there is t ≡ t(r) such that ∀ t ∈ [0, t(r)] u(·, t) ∞,Ω ≤ r.
It follows that for r and t(r) chosen in this way u(·, t) ∞,Ω ≤ r for all t ≤ t(r), i.e., the constructed solution of the regularized problem (4.18) is a weak solution of problem (1.1) in the cylinder Q t(r) . The possibility of continuation of this solution to the maximal interval [0, T * ] follows from the fact that the function u(x, t(r)) possesses the same properties as the initial function u 0 .

Uniqueness theorems
In this section we study the question of uniqueness of weak solutions to the problem The weak solution is understood in the sense of Definition 3.1. Let us assume that the functions a i are continuous with the module of continuity ω, and claim that the function ω is nonnegative and satisfies the condition Without loss of generality we may assume that p + ≥ 2.
We will show that this assumption leads to a contradiction unless µ = 0. Not loosing generality we assume that t = T . Set By the definition of weak solution, for every test-function ζ ∈ Z and τ ∈ [0, T ] Let us denote and write (5.5) in the form Let us introduce the functions depending on the parameters δ ≥ ǫ > 0, and with the function ω(·) defined in (5.3). The definition of F ǫ and (5.4) yield: Letting in (5.6) ζ = F ǫ (w), we obtain: Notice that since δ ≥ ǫ, then Ω δ ⊆ Ω ǫ , |Ω ǫ | ≥ |Ω δ | > µ and, by virtue of (5.3), Let us consider first the case p i ≥ 2. By virtue of (1.3) and the first inequality of (3.24) According to (5.3) Applying Young's inequality, we may estimate the integrand of J in the following way: Let now 1 < p − ≤ p i < 2. Applying (1.3) and the second inequality of (3.24) we have Plugging the pointwise estimates (5.11), (5.12) and (5.13), (5.14) into (5.9) and dropping the nonnegative terms, we arrive at the inequality with a constant C independent of ǫ. Gathering this inequality with (5.10) we obtain the needed contradiction. This means that µ = 0 and w ≤ 0 a.e. in Q T . To complete the proof it suffices to replace u 1 and u 2 .
The above arguments can be extended to non-monotone functions d(z, u). Let We assume that It is easy to calculate that in this case Proposition 5.1. There exists a positive number µ > 2 such that Proof: Set z = s/ǫ and introduce the function Obviously, Since f (z) is monotone increasing for z > 2 and tends to infinity as z → ∞, there is µ ≥ 2 such that f (z) ≥ 0 for z ≥ µ. For z ∈ [1, µ] sF ǫ (s) = z − 1 ≤ µ − 1. Proof: We will adapt the proof of Theorem 5.1. Let u 1 , u 2 be two different solutions of problem (1.1). Set u = u 1 − u 2 . Following the proof of Theorem 5.1 we arrive at the relation with The difference between this case and the one studied in Theorem 5.1 is that now the term I 2 is not sign-defined. By Proposition 5.1 Let us introduce the function Substituting the above inequalities into (5.19) and taking into account (3.24), we find that the function Y (t) satisfies the Gronwall type inequality It follows that Y (t) ≤ K, which contradicts condition (5.10).
Corollary 5.1 (Comparison principle). Let u, v ∈ W(Q T ) be two weak solutions of problem (5.1) such that u(x, 0) ≤ v(x, 0) a.e. in Ω. If the coefficients and the nonlinearity exponents satisfy the conditions of Theorem 5.1 or Theorem 5.2, then u ≤ v a.e. in Q T .

Regularity of solutions for a class of model equations
Let us consider the following simplified version of problem (1.1): We want to trace the dependence of the regularity of weak solutions on the regularity of the data, especially, on the properties of the exponents p i (z) and σ(z). Let us accept the notations Then the weak solution of problem (6.1) satisfies the estimate with an absolute constant C = C(p ± , σ ± , q, T, K).
Proof: Let us recall that under the conditions of Theorem 6.1 a weak solution of problem (6.1) can be obtained as the limit of the sequence of Galerkin's approximations (see the proof of Theorem 3.1) where the system {ψ k } is dense in W 1,p + 0 (Ω), and the functions u For the sake of simplicity, throughout the proof we use the notation u for the approximate solution u (m) . Fix some m, multiply relations (3.5) by u (m) k,t and take the sum over k = 1, . . . , m. This gives the equality We will use the easily verified formulas Gathering these estimate we obtain the inequality The terms I 3,i are estimated likewise: It follows that (c|u| σ ln |u| σ |σ t |) dx + C (λ c (t) + λ σ (t)) Y (t) + Cλ σ (t).
Remark 6.1. The assertion remains true for the solutions of the equation provided that the functions c i (z), σ i (z) satisfy the conditions of Theorem 6.1.
Remark 6.2. Problem (6.1) includes, as a partial case, the problem (6.10) u t = ∆ p u + f in Q T , u = 0 on Γ T , u(x, 0) = u 0 (x) in Ω, with constant p ∈ (1, ∞). It was proved in [19, Lemma 2.1] that the solutions of this problem satisfy the estimate (6.11) u t 2 L 2 (QT ) + u p L ∞ (0,T ;W 1,p (Ω)) ≤ C f 2 L 2 (QT ) + u 0 p W 1,p (Ω) , which is contained in (6.3) if p = const. Moreover, if in the conditions of Theorem 6.1 the coefficients a i , c and the exponents p i and σ are variable but independent of t, then estimate (6.11) is true as well for the solutions of problem (6.1).

Extensions
The results of this paper can be extended in various directions. Let us mention here several most obvious generalizations. The rest of the arguments does not need any change.
4. The proofs of the existence theorems can be adapted to the case of the Neumann boundary condition. For example, let us consider the problem (7.1) where ν = (ν 1 , . . . , ν n ) denotes the outer normal vector to Γ T . Let us introduce the function spaces We say that function u(x, t) ∈ W(Q T ) ∩ L ∞ (0, T ; L 2 (Ω)) is a weak solution of problem (7.1) if for every test-function ζ ∈ Z ≡ {η(z) : η ∈ W(Q T ) ∩ L ∞ 0, T ; L 2 (Ω) , η t ∈ W ′ (Q T )}, and every t 1 , t 2 ∈ [0, T ] the following identity holds: Let us assume that the data of problem (7.1) satisfy the conditions of Theorem 3.1. Since the space W(Q T ) is separable, a solution of problem (7.1) can be constructed as the limit of the sequence of Galerkin's approximations (see the proof of Theorem 3.1).