Local Boundedness of Weak Solutions for Nonlinear Parabolic Problem with -Growth

Abstract

We study the nonlinear parabolic problem with -growth conditions in the space and give a local boundedness theorem of weak solutions for the following equation , where and satisfy -growth conditions with respect to and .

1. Introduction

The study of variational problems with nonstandard growth conditions is an interesting topic in recent years. -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 example [1–9].

Let be , where is given. In [8], the authors studied the following equation:

(11)

where , is dependent on the space variable and the time variable , is the local weak solution in the space , and the authors proved the local boundedness of the local weak solution in . In this paper, we will study the following more general problem:

(12)

(13)

(14)

where is a given function in and is an elliptic operator of the form with the coefficients and satisfying the classical Leray-Lions conditions. In [10], we have proved the existence of the solutions of (1.2)–(1.4) and have gotten ; in this paper we will give the local boundedness theorem of the weak solutions in the framework space , which can be considered as a special case of the space .

Many authors have already studied the boundedness of weak solutions of parabolic equation with -growth conditions, where is a constant, for example [8, 11–15]. The boundedness of the weak solutions plays a central role in many aspects. Based on the boundedness, we can further study the regularity of the solutions. For example, first in [15] the author studied the equation

(15)

and got -estimates of the degenerate parabolic equation with -growth conditions for , where is a constant, then in [16] the authors established the Hölder continuity of the equation for the singular case , and in [17] the authors discussed Harnack estimates for the bounded solutions of the above parabolic equation for .

The space provides a suitable framework to discuss some physical problems. In [18], the authors studied a functional with variable exponent, , which provided a model for image denoising, enhancement, and restoration. Because in [18] the direction and speed of diffusion at each location depended on the local behavior, only depended on the location in the image. Consider that the space was introduced and discussed in [10] and [19], we think that the space is a reasonable framework to discuss the -growth problem (1.2)–(1.4), where only depends on the space variable similar to [18].

In this paper, let and be the operators such that for any and , and are both continuous in for a.e. and measurable in for all . They also satisfy that for a.e. , any and :

(16)

(17)

(18)

(19)

where are constants.

Throughout this paper, unless special statement, we always suppose that is -continuous on , that is, for every , and satisfy

(110)

is the conjugate function of .

Definition 1.1.

A function is called a weak solution of (1.2)–(1.4) if

(111)

for all .

We will prove the following local boundedness theorem.

Theorem 1.2.

Let . If is a nonnegative local weak solution of (1.2)–(1.4), then is locally bounded in Q. Moreover, there exists a constant such that for any and any ,

(112)

where for all , , , , and .

2. Preliminaries

We first recall some facts on spaces , , and . For the details, see [19–21].

Although we assume (1.10) holds in this paper, in this section we introduce the general spaces , , and .

Denote

(21)

where is an open subset.

Let be an element in . Denote . For , we define

(22)

The space is

(23)

endowed with the norm

(24)

We define the conjugate function of by

(25)

Lemma 2.1 (see [21]).

• (1)  The dual space of is if .

• (2)  The space is reflexive if and only if (1.10) is satisfied.

Lemma 2.2 (see [21]).

If , is dense in the space and is separable.

Lemma 2.3 (see [21]).

Let , for every and , we have

(26)

where C is only dependent on and , not dependent on .

Next let be an integer. For each , are nonnegative integers and , and denote by the distributional derivative of order with respect to the variable .

We now introduce the generalized Lebesgue-Sobolev space which is defined as

(27)

is a Banach space endowed with the norm

(28)

The space is defined as the closure of in . The dual space is denoted by equipped with the norm

(29)

where infimum is taken on all possible decompositions

(210)

Lemma 2.4 (see [21]).

(1)   and are separable if .

(2)   and are reflexive if (1.10) holds.

We define the space as the following:

(211)

is a Banach space with the norm , where is independent of .

The space is defined as the closure of in , and is continuous embedding. Let be the number of multiindexes which satisfies , then the space can be considered as a close subspace of the product space . So if , is reflexive and further we can get that the space is reflexive. The dual space is denoted by equipped with the norm

(212)

where infimum is taken on all possible decompositions

(213)

Next, we will introduce some results in [22].

Lemma 2.5.

Let be a sequence of positive numbers, satisfying the inequalities , where and are given numbers. If , then converges to 0 as .

Lemma 2.6.

There exists a constant depending only on , such that for every ,

(214)

where .

Remark 2.7.

In [10], we have gotten that for the Galerkin solutions , strongly in , weakly in , weakly in and weakly in .

3. Proof of the Theorem

Suppose that is a weak solution of (1.2)–(1.4), then there exists such that

(31)

Indeed, by Young's inequality, we have

(32)

where is the Lebesgue measure of . Since and , we can get . Then by Lemma 2.6, we get

(33)

where . Thus the desired result is obtained.

We define . Fix a point in . Let , , and . Fix and consider the sequences

(34)

and the corresponding cylinders . It follows from the definitions that

(35)

We consider also the boxes , where for

(36)

For these boxes, we have the inclusion

(37)

We introduce the sequence of increasing levels

(38)

Let be the Galerkin solutions in [10]. Similarly, we can get is bounded in . Since converges to 0 in , by interpolation inequality, we have

(39)

where , . Furthermore, strongly in . Since , strongly in . In the same way, we obtain that strongly in ; furthermore, we get for a.e. .

Let and be the smooth cutoff function satisfying

(310)

Take as the testing function in the following equation:

(311)

First, by for a.e. and strongly in , we get

(312)

By Fatou's lemma, we get

(313)

Because strongly in and weakly in , we get

(314)

Since strongly in and weakly in , we have

(315)

Then for the remaining parts of (3.11), we get

(316)

By (1.6), (1.7), and (1.9),

(317)

As , by Young's inequality and Hölder's inequality, we have

(318)

In the same way, by and Young's inequality, we have

(319)

For a set , meas is the Lebesgue measure of . Let and . By (3.11)–(3.19), we get

(320)

Moreover, we observe that for to be determined later,

(321)

thus we get

(322)

Then for and in (3.22), by Hölder inequality, we obtain respectively

(323)

(324)

For the integral involving , first we write , then we obtain

(325)

By Young's inequality and (3.25), we get

(326)

Let , then . By (3.20)–(3.24) and (3.26), we obtain

(327)

By Young's inequality,

(328)

Moreover, by (3.27), we can get

(329)

Next we define the smooth cutoff function in

(330)

For the function , by Lemma 2.6 and (3.29), we get

(331)

Finally, we define , Let ; by Hölder inequality, we obtain

(332)

where . Then by Lemma 2.5, we have as , provided is chosen to satisfy

(333)

By , we can get as . Since and a.e. in , by Lebesuge's theorem we get . So we obtain a.e. in .

Thus we get

(334)

Remark 3.1.

In this paper, we study the boundedness of weak solution in the case . For the singular case , the conditions in the paper are not enough. In [22], there is a counterexample in 13 of Chapter XII. The author studied the solutions of the homogeneous equation

(335)

where

(336)

and proved that the solution is unbounded in .

Remark 3.2.

In general, we consider the equation

(337)

where

(338)

and is an elliptic operator of the form . and satisfy that for a.e. , any and :

(339)

where , , and are constants.

Similarly, we can get the following theorem.

Theorem 3.3.

Let . If is a nonnegative local weak solution of (3.37), (1.3), and (1.4), then is locally bounded in Q. Moreover, there exists a constant such that for any and any ,

(340)

where for all , , , , and , .