Abstract

We deal with existence results for nonlinear parabolic equations with general quadratic gradient terms and with absorption term which depend on the solution. We note that no boundedness is assumed on the data of the problem. We prove an existence result of distributional solution via test-function method. A priori estimates and compactness arguments are our main ingredient; the method of sub-supersolution does not apply her.

1. Introduction

This work is devoted to deal with existence results concerning nonlinear parabolic equations with both first-order terms having quadratic growth with respect to the gradients and superlinear absorption terms which depend on the solution. Let us consider the following Cauchy-Dirichlet problem: where is a pseudomonotone, coercive, uniformly elliptic operator acting from to its dual. is a bounded open set in , and . The operator grows like . The unknown real function depends on and . Let be a Carathéodory function (i.e., is measurable in for every in ).(A₁)There exists a constant such that for almost every in and every in , where is a nonnegative function in .(A₂)There exists a constant such that for almost every in and every in .(A₃) 

for almost every in , for every in , for every in .

Let us define the differential operator as follows:

The function satisfies The initial data satisfy the following hypothesis: where

The source term satisfies the same condition considered by Aronson and Serrin in [1] to show the existence of a solution for the classical problem . The condition on the source term is optimal. Indeed, if or if , with , the condition (1.6) is still satisfied.

Let us note that we studied the elliptic problem associated to in [2]. This kind of problems has been extensively studied in the last years by many authors (see, e.g., [3–13] and the references therein). In these works, the hypothesis on the function implies grosso modo that is bounded, with some restrictions as in [4, 10, 14]. A special condition is assumed in [10], where is supposed to tend to for tending to .

There is many results for particular situations of our problem (see, e.g., [4, 7, 12, 13]). All this work have studied the question of existence of distributional solutions for this problem in the case where , , and with The existence has also been studied in the case where and , , which are special cases of our conditions.

In the case where , and for more general condition on , existence results of a solution for parabolic convection diffusion problems have been given in [15], [8].

We can reduce this problem with the change of with the Col-Hopf change of variable to the following

In this stage, we have an existence result of distributional solutions via test function method. That gives the a priori estimates for the approximate problem associated with (1.11) which also provide a priori estimates for the approximate problem associated with (1.9) and, therefore, an existence result of distributional solution for problem (1.9).

One cannot perform such a change of variable, when trying to extend the previous results to our more general situation, where one has a general first-order term which grows quadratically with respect to the gradient and with superlinear reaction terms which grow like . Therefore, we shall use some convenient test functions to prove the a priori estimates and use compactness arguments to prove an existence result of distributional solutions of . We point out that for this class of problems, the regularity assumed on the data and , can not expect bounded solutions.

We also point out that we are interested in solutions having finite norms in . The techniques used in this paper are mainly based on a linear operator and on the concept of distributional solutions. These approachs allow to have, in the case of both subcritical growth and a reaction terms with , existence results. The first ingredient of our proof consists in obtaining certain a priori bounds on the solutions of approximate problems and some suitable -norm of diffusion terms. A convenient use of Young's inequality will give a uniform estimate of the -norms and, therefore, the weak convergence up to a subsequence. We will prove that there exists such that, up to a subsequence the solution of the approximate problems converges to , all everywhere convergence of gradient of to gradient of , up to a subsequence, which is important in the study of the limiting process. Next, we will prove the convergence of the superlinear reaction term and the quadratic gradient term in . Another interesting approach is in some sense the combination of the previous, in studies of the behavior of sequences of approximating solutions. Likewise, we will see that the solutions of the approximates problems converge to the solution of the model problem in , which gives meaning to the initial condition.

2. Basic Results

Let be a bounded domain in , . We denote by for the set and by the set .

We consider the following nonlinear problem that we denote by where the unknown function is a real function depending on and . and are differential operators such that

Let us consider the following assumptions.(H₁)The real function is such that (H₂)The real functions and are satisfying (1.7) and (1.6), respectively

The operator is such that where(H₃) and is a real function such that

By a weak solution of the problem (1.1), we mean a function such that and satisfying for any test function in (the functions with compact support).

In the sequel we denote by a truncation function satisfying , , for , for and , where is a positive real. By we denote different constants in which may vary from line to line. The main result in this paper is the following.

Theorem 2.1. If the hypotheses – and – are satisfied, then the problem (1.1) admits at least one solution , such that

To prove the main result, we approximate our problem by a sequence of regular problems and show a priori estimates of solutions. Next, we shall prove the convergence of approximating solutions to some function that solves our problem.

Let us recall the classical inequality of Poincaré and Sobolev (see [16, Chapters 7.7 and 7.8]).

Lemma 2.2. Let be an open subset of with finite Lebesgue measure. Then, for every such that , one has the following inequality: where is the measure of the unit ball in . Furthermore, there exists a constant such that, , where .

Remark 2.3. By an approximation argument, the same inequality holds true if we replace by .

Let us recall next the Gagliardo-Nirenberg's inequality for evolution spaces.

Lemma 2.4 (see, e.g., [17]). Let be a bounded open set of and a real positive number. Let be a function such that . Then and the following estimate holds

We are interested in studding a sequence of regular problems approximating the model problem. We prove the existence of bounded solutions for the approximating problems, and this bound does not depend on . We shall prove some a priori estimates on the solutions of this sequence of problems which serves in the limiting process.

3. Approximating Problems

We regularize the problem (1.1) by considering the following sequence of problems: where Let us consider

Next, we consider the truncated function

We denote by a strictly increasing sequence of bounded sets invading . Next we denote

From standard results (see, e.g., [14]), the following problem: where admits at least one solution satisfying

Then one has the following estimates:

Indeed, let us define the following function: wehere

We introduce the function defined by Let us consider the following sequences:

Taking with , we obtain

Applying Young's inequality, one has the following inequality:

We now choose such that , which is possible, since and . We use again Young's inequality twice and we obtain Then

From (3.14), we obtain

In consequence,

Next, we substitute by for any , in (3.18), which is possible. We obtain where is a constant that does not depend on . then, the approximate problem admits at least one solution which is bounded independently on in .

Let us now prove that the sequence is bounded in . We denote where

We define, for any fixed, the functions defined as follows: Let us consider the following sequences:

We can choose as test function, and we obtain Then, we deduce In consequence,

Finally, we obtain Then, the sequence is bounded in .

We require the all everywhere convergence of gradient to the gradient of . Let us consider

Substituting in the approximating problem successively with and , we consider the following function: After substraction, for , , we obtain the following inequality: Let us now consider for sufficiently large the following sequence: Then we get Since (3.20) holds, then is in . So that, is bounded in , then one has

Using Holder's inequality, we obtain

Using (3.34), one has

On the other hand, since the measure of converges to 0 as , then, as converges to , then

Therefore

4. Limiting Process

We denote by the solution of the approximate problems on with initial condition . To prove the main result, we deal with the limiting process of the approximating problems. First of all, we will prove that there exists such that, up to a subsequence, converges to , for almost every . First, we will prove the all everywhere convergence of the gradients of to the gradient of , up to a subsequence, in . Next, we will prove the convergence of the superlinear reaction term and the quadratic gradient term in . Finally, we will see that converges to in , which gives meaning to the initial condition.

From (3.38) and up to a subsequence , we have

By consequence, since is bounded in , Vitali's theorem implies that

Since one has

By a diagonal process, we may select a subsequence, also denoted by , such that and also

From the construction of and , we have Since , Then using compactness arguments (see [18]), we have

From (4.5) and the fact that is bounded in , the equi-integrability of is derived and then from Vitali's theorem, we have By consequence,

Since then Therefore, from (4.1) and Vitali's theorem, we conclude that

Finally, the sequence belongs to and ; this implies that the initial condition is satisfied.