Bounded Solutions to Nonlinear Parabolic Equations
Jaouad Igbida1
Academic Editor: S. Zhang
Received08 Jul 2011
Accepted11 Aug 2011
Published13 Oct 2011
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.
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., [3β13] 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 β and β 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.
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
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
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.
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.
References
D. G. Aronson and J. Serrin, βLocal behavior of solutions of quasilinear parabolic equations,β Archive for Rational Mechanics and Analysis, vol. 25, pp. 81β122, 1967.
A. El Hachimi and J. Igbida, βBounded weak solutions to nonlinear elliptic equations,β Electronic Journal of Qualitative Theory of Differential Equations, no. 10, pp. 1β16, 2009.
M. Ben-Artzi, P. Souplet, and F. B. Weissler, βThe local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces,β Journal de Mathématiques Pures et Appliquées, vol. 81, no. 4, pp. 343β378, 2002.
L. Boccardo, F. Murat, and J.-P. Puel, βExistence results for some quasilinear parabolic equations,β Nonlinear Analysis. Theory, Methods & Applications, vol. 13, no. 4, pp. 373β392, 1989.
D. Blanchard and A. Porretta, βNonlinear parabolic equations with natural growth terms and measure initial data,β Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, vol. 30, no. 3-4, pp. 583β622, 2001.
L. Boccardo and M. M. Porzio, βBounded solutions for a class of quasi-linear parabolic problems with a quadratic gradient term,β in Evolution Equations, Semigroups and Functional Analysis, vol. 50 of Progress in Nonlinear Differential Equations and Their Applications, pp. 39β48, Birkhäuser, Basel, Switzerland, 2002.
A. Dall'aglio, D. Giachetti, and J.-P. Puel, βNonlinear parabolic equations with natural growth in general domains,β Bollettino della Unione Matematica Italiana, 2004.
A. Dall'Aglio, D. Giachetti, C. Leone, and S. S. de León, βQuasi-linear parabolic equations with degenerate coercivity having a quadratic gradient term,β Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 23, no. 1, pp. 97β126, 2006.
A. Dall'Aglio and L. Orsina, βNonlinear parabolic equations with natural growth conditions and data,β Nonlinear Analysis. Theory, Methods & Applications, vol. 27, no. 1, pp. 59β73, 1996.
V. Ferone, M. R. Posteraro, and J. M. Rakotoson, βNonlinear parabolic problems with critical growth and unbounded data,β Indiana University Mathematics Journal, vol. 50, no. 3, pp. 1201β1215, 2001.
R. Landes and V. Mustonen, βOn parabolic initial-boundary value problems with critical growth for the gradient,β Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 11, no. 2, pp. 135β158, 1994.
A. Mokrane, βExistence of bounded solutions of some nonlinear parabolic equations,β Proceedings of the Royal Society of Edinburgh, vol. 107, no. 3-4, pp. 313β326, 1987.
L. Orsina and M. M. Porzio, β-estimate and existence of solutions for some nonlinear parabolic equations,β Unione Matematica Italiana B, vol. 6, no. 3, pp. 631β647, 1992.
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, USA, 1968.
A. Dall’Aglio, D. Giachetti, and C. Leone, βSemilinear parabolic equations with superlinear reaction terms, and application to some convection-diffusion problems,β Ukrainian Mathematical Bulletin, 2004.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 2nd edition, 1983.
E. DiBenedetto, Degenerate Parabolic Equations, Springer, New York, NY, USA, 1993.
J. Simon, βCompact sets in the space ,β Annali di Matematica Pura ed Applicata, vol. 146, pp. 65β96, 1987.