Regularity of Weak Solutions for Nonlinear Parabolic Problem with p(x)-Growth

In this paper, we study the nonlinear parabolic problem with p(x)growth conditions in the space W 1,x L p(x) (Q), and give a regularity theorem of weak solutions for the following equation


Introduction
In recent years, the research of variational problems with nonstandard growth conditions is an interesting topic. p(x)-growth problems can be regarded as a kind of nonstandard growth problems and they appear in nonlinear elastic, electrorheological fluids and other physics phenomena. Many results have been obtained on this kind of problems, for examples [1][2][3][4][5][6][7][8][9].
In this paper, we will qualitatively study the properties of weak solutions. For more information about qualitative analysis, we refer to [10][11]. Let Q be Ω × (0, T ) where T > 0 is given. In [8], the authors studied the following equation in the space W 1,p(x,t) loc (Q) ∩ C(0, T ; L 2 loc (Ω)), p(x, t) = p 2 < ∞, p(x, t) is dependent on the space variable x and the time variable t, and satisfies the following Logarithmic Hölder condition |p(x, t) − p(y, s)| ≤ C 1 − ln(|x − y| + C 2 |t − s| p2 ) for all (x, t), (y, s) ∈ Q, |x − y| < 1 2 , |t − s| < 1 2 , where C 1 , C 1 > 0 are constants. The authors proved the Hölder continuity of the local weak solution with the scale transformation method.In this paper, we will study the following more general problem ∂u ∂t + A(u) = 0, in Q, (1.1) u(x, t) = 0, on ∂Ω × (0, T ), (1.2) u(x, 0) = ψ(x), in Ω, (1.3) where ψ(x) is a given function in L 2 (Ω) and A : W 1,x 0 L p(x) (Q) → W −1,x L q(x) (Q) is an elliptic operator of the form A(u) = −diva(x, t, u, ∇u)+a 0 (x, t, u, ∇u) with the coefficients a and a 0 satisfying the classical Leray-Lions conditions. In [12][13] we have proved the existence and the local boundedness of the solutions of (1.1)-(1.3) and have obtained u ∈ W 1,x L p(x) (Q) ∩ L ∞ (0, T ; L 2 (Ω)). In this paper we will give the regularity theorem of the weak solutions in the framework space W 1,x L p(x) (Q), which can be considered as a special case of the space W 1,p(x,t) (Q).
The space W 1,x L p(x) (Q) provides a suitable framework to discuss some physical problems. In [14], the authors studied a functional with variable exponent, 1 ≤ p(x) ≤ 2, which provided a model for image denoising, enhancement, and restoration. Because in [14] the direction and speed of diffusion at each location depended on the local behavior, p(x) only depended on the location x in the image. Consider that the space W 1,x L p(x) (Q) was introduced and discussed in [12] and [15], we think that the space W 1,x L p(x) (Q) is a reasonable framework to discuss the p(x)-growth problem (1.1)-(1.3), where p(x) only depends on the space variable x similar to [14].
We define the conjugate function q(x) of p(x) by where C is only dependent on p(x) and Ω, not dependent on u(x), v(x).
We now introduce the generalized Lebesgue-Sobolev space W m,p(x) (Ω) which is defined as where infimum is taken on all possible decompositions Lemma 2.4(see [18] (Ω) are reflexive if (1.9) holds. We define the space W m,x L p(x) (Q) as the following: where infimum is taken on all possible decompositions Next, we will introduce the parabolic space and some results in [16]: Definition 2.6 Let p, r ≥ 1. We define the function spaces Lemma 2.7 let {Y n }, n = 0, 1, 2, · · · , be a sequence of positive numbers, satisfying the inequalities Y n+1 ≤ Cb n Y 1+α n , where C, b > 1 and α > 0 are given numbers.
) for some ρ > 0 and some x 0 ∈ R N , and let k and h be any pair of real numbers such that k < h, then there exists a constant C depending only upon N , such that Similarly, we can get the following lemma in variable exponent space.
Proof: Because p(x) is bounded and independent of t. We only need to notice that there exist u k ∈ C 1 0 (Q) such that u k → u in L p(x) (Q), and by the uniform continuity of u k , we can conclude the lemma.2

Regularity of Weak Solutions
In [12][13] , we have obtained that for the Galerkin solution u n ∈ C 1 (0, T ; C ∞ 0 (Ω)), u n → u strongly in L 2 (Q) and L p(x) (Q), u n ⇀ u weakly in W 1,x 0 L p(x) (Q), a(x, t, u n , ∇u n ) ⇀ a(x, t, u, ∇u) and a 0 (x, t, u n , ∇u n ) ⇀ a 0 (x, t, u, ∇u) weakly in L q(x) (Q), u n → u a.e. in Q and ∇u n → ∇u a.e. in Q.
For (1.11), integrating by parts, we can get As a(x, t, u n , ∇u n ) ⇀ a(x, t, u, ∇u) weakly in L q(x) (Q) and a 0 ( In (3.1), let ϕ be independent of t and t = t + h, then we get then for the sufficient large m, we have and lim namely {v n } is a Cauchy sequence in C(0, T ; L 2 (Ω)), so we get the result.2 Next, we will prove the main theorem. By [13], we know that there exists a constant M > 0, such that u L ∞ loc (Q) ≤ M . Fix a point (x 0 , t 0 ) in Q, let ρ ∈ (0, 1) be small enough such that p(x).
Denote µ + = ess sup where A > 2 is a constant to be determined later. We assume that where ε ∈ (0, 1) will be determined later. This implies the inclusion and ess osc Take C = A, then the first iterative of proposition 3.4 is hold, so the proposition 3.4 is right. therefore we also assume that (3.3) is hold in the following proof. Let We assume (x 0 , t 0 ) = (0, 0) and define (u − k) ± = max{±(u − k), 0}. Lemma 3.2 There exists a number σ ∈ (0, 1) independent of ω, ρ such that if (3.3) and Proof: Up to a translation we may assume that (0, t * ) = (0, 0). Let .
Since u n → u in L 2 (Q) and u ∈ C(0, T ; L 2 (Ω)), u n → u in L 2 (Ω) for ∀t ∈ (0, T ), therefore we can get Since ∇(u n − k m ) − → ∇(u − k m ) − and a(x, t, u n , ∇u n ) → a(x, t, u, ∇u) a.e. in Q t m , by Fatou lemma, By the fact that u n → u strongly in L p(x) (Q), a(x, t, u n , ∇u n ) ⇀ a(x, t, u, ∇u) weakly and a 0 (x, t, u n , ∇u n ) ⇀ a 0 (x, t, u, ∇u) weakly in L q(x) (Q), we have where C = C(M, p + ).
On the other hand, we have By lemma 2.8, ≡ σ which just satisfies the condition of this lemma, i.e.