UNIQUENESS AND COMPARISON THEOREMS FOR SOLUTIONS OF DOUBLY NONLINEAR PARABOLIC EQUATIONS WITH NONSTANDARD GROWTH CONDITIONS

The paper addresses the Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: ut = div ( a(x, t, u)|u|α(x,t)|∇u|p(x,t)−2∇u ) + f(x, t) with given variable exponents α(x, t) and p(x, t). We establish conditions on the data which guarantee the comparison principle and uniqueness of bounded weak solutions in suitable function spaces of Orlicz-Sobolev type.

Problem (3) will be the subject of the further study. Equations of the types (1) and (3) with constant exponents α and p arise in the mathematical modelling of various physical processes such as flows of incompressible turbulent fluids or gases in pipes, processes of filtration in porous media, glaciology -see [5,6,16,17,22,33] and further references therein. The questions of existence and uniqueness of solutions to equations like (1) and (3) with constant exponents of nonlinearity α and p were studied by many authors -see [6,14,15,16,24,28,29] for equations of the type (1) and [17,21] for the equations of the type (3) with the prescribed function B ≡ B(x, t) independent of the solution v. Existence, uniqueness, and qualitative properties of solutions for parabolic equations with variable nonlinearity corresponding to the special cases α(x, t) = 0, or p(x, t) = 2 were studied in [1,2,3,4,8,9,10], see also [7] for a review of results concerning elliptic equations with variable nonlinearity. The Cauchy problem for doubly nonlinear parabolic equations with constant exponents of nonlinearities is studied in [30,31,32]. The theorem of existence of weak solution to problem (3) (and correspondingly problem (1)) is proved in [11]. Existence of bounded solutions for elliptic equations of this type is proved in [12].
In the present work we prove comparison principle and uniqueness of weak solutions for the Dirichlet problem (3) in which the exponents α and p are allowed to be variable. Also we consider localization properties of weak solutions.
The paper is organized as follows. In Section 2 we prove several auxiliary assertions and collect some known facts from the theory of Orlicz-Sobolev spaces. The precise assumptions on the data and main results are given in Section 3. In Section 4 we derive formulas of integration by parts. In Sections 5, 6 we give the proofs of the main comparison theorems. The comparison principle and uniqueness are proved for the solutions subject to some additional restrictions, but under weaker assumptions on the data, and is independent of the proof of the existence theorem. To be precise, the comparison principle and uniqueness are true for the weak solutions with ∂ t Φ 0 (z, v) ∈ L 1 (Q). In order to ensure that this class of solutions is nonempty, in the final Section 7 we show that the already constructed solution belongs to this class, provided that the data of the problem satisfy some additional conditions.

2.2.
Parabolic spaces L p(·,·) (Q) and W(Q). Let p(z), z = (x, t) ∈ Q, satisfy condition (4) in the cylinder Q. For every fixed t ∈ [0, T ] we introduce the Banach space and denote by V t (Ω) its dual. By W(Q) we denote the Banach space W (Q) is the dual of W(Q) (the space of linear functionals over W(Q)): Since V + (Ω) is separable, it is a span of a countable set of linearly independent functions {ψ k (x)} ⊂ V + (Ω). We will need two elementary inequalities.
which is formally equivalent to problem (1). Throughout the paper we assume that the coefficient a(z, r) and the exponents on nonlinearity p(z), α(z) satisfy the following conditions: • a(z, r) is a Carathéodory function such that there exists a ± such that The solution of problem (10) is understood in the following sense.
The main existence result is given in the following theorem.
Let conditions (11), (12) be fulfilled. Then for every weak solutions v 1 , v 2 , such that The uniqueness is proved in a narrower class of functions than the existence, but since the proofs of Theorems 3.3, 3.4 are practically independent on the proof of Theorem 3.2, the conditions on the exponents α(z), p(z) are less restrictive. For the sake of completeness of presentation, in the end of the paper we present the conditions on the data of problem (10) which guarantee that the corresponding solution satisfy the conditions of the comparison and uniqueness theorems.
4. Formulas of integration by parts. Let ρ be the Friedrich's 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 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 In the same way we check that By the Lebesgue differentiation theorem Proof. Let u h ∈ C ∞ (Q) be the mollification of u ∈ W(Q) and Since u and u h are bounded by a constant 1 + K 0 , and γ(z) ≥ γ − > −1, it follows from Propositions 1, 2 that Explicitly calculating the primitive, in the same way we check that for every s > 1 Let ψ k (z) = χ k (t) γ + 2 with the function χ k introduced in (15). Following the proof of Lemma 4.3, we find: Since u ∈ W(Q) ∩ L ∞ (Q) and γ − > −1, v ∈ W(Q) for every > 0. Indeed: since u L ∞ (Q) ≤ M , we have the estimates which provide the inclusion |∇v| ≤ ( + |u|) γ(z) |∇u| + |∇γ| |u| 0 ( + |s|) γ(z) | ln ( + |s|)| ds ∈ L p(·) (Q).
We may now pass to the limit as h → 0 in every term of (17), following the proof of Lemma 4.3: Letting k → ∞ and applying the Lebesgue differentiation theorem, we arrive at (16).
Under the foregoing conditions on the exponents p(z) and γ(z) the following formula of integration by parts holds: ∀ a.e.t 1 , Let us introduce the function space with Φ 0 defined in (2) and define the functions with It is easy to see that For a.e. θ ∈ (0, T ) there exists the limit Proof. Since w ∈ L ∞ (Q) and φ = φ k,δ,θ are uniformly bounded, it follows from the dominated convergence theorem that and, because sign v = sign w, On the other hand, repeating the same arguments with the test-function φ k,δ,θ ≡ χ k,θ (t) T δ (w), we find that The straightforward computation shows that Letting k → ∞, δ → 0 and applying the Lebesgue differentiation theorem, we find that for a.e. θ ∈ (0, T ) By (14) for every test-function φ ∈ W(Q) Taking for the test-function φ k,δ,θ defined in (18) and applying Lemma 4.5 we have that for a.e. θ ∈ (0, T ) there exists the limit of the first term on the left-hand side of (19): The second term on the left-hand side of (19) with φ(z) = χ k,θ (t) T δ (v(z)) is represented in the form Let us denote (3)). Passing to the limit as k → ∞, for every fixed δ and θ we obtain the equality Making use of the well-known inequality dz. Next, To estimate J (1) (δ) we make use of the following elementary lemma.
Proof. The assertion follows from Young's inequality Applying Lemma 5.1 we have: By Young's inequality Gathering (21), (22) and (23) we arrive at the inequality Choosing ≡ (p − ) sufficiently small we then have with a positive constant C ≡ C(p ± ). It remains to show that the right-hand side of the last inequality tends to zero as δ → 0. We will use the following Lemmas.

7.
Existence of solutions u ∈ V(Q): L 1 -estimate for ∂ t Φ(z, v). Let us check that problem (10) indeed admits solutions in V(Q), which means that the class of uniqueness is nonempty. Following [11], we construct a solution as the limit of the sequence of solutions of the regularized problems with the coefficient depending on the given parameters > 0, K > 0. For every ∈ (0, 1) and 1 < K < ∞ the coefficient A ,K (z, u ) is separated away from zero and infinity, so that problem (27) can be regarded as the Dirichlet problem for the evolutional p(z)-Laplacian.
Theorem 7.1 ( [10]). For every u 0 ∈ L 2 (Ω), f ∈ L 2 (Q), > 0, K > 0 problem (27) has at least one weak solution u ∈ L ∞ (0, T ; L 2 (Ω))∩W(Q) such that ∂ t u ∈ W (Q) and for every test-function φ ∈ L ∞ (0, T ; Moreover, if u 0 ∈ L ∞ (Ω), f ∈ L 1 (0, T ; L ∞ (Ω)), this solution belongs to L ∞ (Q) and obeys the estimate As a byproduct we also have that for every φ ∈ W(Q) (see [10]) The solution of problem (27) is obtained as the limit as m → ∞ of the sequence of Galerkin's approximations, where the family {ψ i (x)} is dense in V + (Ω) and forms an orthogonal basis of L 2 (Ω). Estimate (28) makes the coefficient A ,K (z, u ) independent of K, provided that K ≥ K 0 + 1: Problem (27) is considered then as a problem with the unique regularization parameter . Passage to the limit as → 0 is justified in [11,Sec.5] in the proof of Theorem 3.1. To this end problem (27) is substituted by the formally equivalent problem and The proof is based on the uniform a priori estimates for the functions v , ∇v and ∇v + B(v ) in the variable Lebesgue spaces L p(z) (Q), the integration-by-parts formulas (see Lemma 4.4), and the monotonicity of the elliptic part of equation (31). The proof of integrability of ∂ t Φ 0 (z, v) ≡ ∂ t u is thus reduced to checking that for the solutions v (m) of the regularized problems (31) the norms ∂ t Φ (z, v (m) ) 1,Q are bounded uniformly with respect to and m. By virtue of (30) and (32), the coefficients c i,m, (t) are defined as the solutions of the system of the ordinary nonlinear differential equations where u 0i and f i (t) are the Fourier coefficients of the functions u 0 (x) and f (z) in the basis {ψ i }: The function u (m) = Φ (z, v (m) ) defined by (30) is a weak solution of problem (31) with the data u (m) 0 , f (m) and satisfies (29) with an arbitrary φ ∈ W(Q). Let us fix some > 0, m ∈ N, and introduce the function in t, we write the equation for V in the form The straightforward calculation gives the equalities Combining these formulas we conclude that Let us introduce the functions According to the definition h µ (σ) ≥ 0, lim η→0 σh µ (σ) = 0, |H µ (σ)| ≤ 1, lim η→0 H µ (σ) = sign σ, lim µ→0 H µ (σ) = |σ| .
Multiplying (33) by H µ (V ) and integrating by parts in t, we arrive at the equality Let us consider the simple case: p t = 0, γ t = 0, Φ ≡ Φ (x, v). In this case the previous equality becomes Let us write (35) in the form with Dropping the nonpositive term I 1 on the right-hand side of (36), letting µ → 0 and using (34) we finally obtain: Since the right-hand side of this inequality is independent of m and , the needed estimate follows by passing to the limit as m → ∞ and → 0.