Bounded Solutions to Nonlinear Parabolic Equations

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.


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 Λu − div a x, t, u, ∇u is a pseudomonotone, coercive, uniformly elliptic operator acting from L 2 0, T; H 1 0 Ω to its dual.Ω is a bounded open set in R N , N ≥ 1 and T > 0. The operator A x, u grows like |u| r .The unknown real function u depends on x ∈ Ω and t ∈ 0, T .Let a : Let us define the differential operator A x, u as follows: A x, u a x u|u| r−1 .

1.5
The function f satisfies The initial data satisfy the following hypothesis: The source term f satisfies the same condition considered by Aronson and Serrin in 1 to show the existence of a solution for the classical problem ∂u/∂t − Δu f x, t .The condition on the source term f is optimal.Indeed, if f ∈ L ∞ 0, T; L q Ω or if f x, t f x ∈ L q Ω , with q > N/2 , the condition 1.6 is still satisfied.
Let us note that we studied the elliptic problem associated to 1.1 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 β implies grosso modo that β is bounded, with some restrictions as in 4, 10, 14 .A special condition is assumed in 10 , where β is supposed to tend to ∞ for s 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 a ≡ 0, u 0 ∈ L ∞ Ω , and f x, t ∈ L r 0, T; L q Ω with q r − 1 /r > N/2 The existence has also been studied in the case where u 0 ∈ L ∞ Ω and f x, t f x ∈ L m Ω× 0, T , m > 1 N/2 , which are special cases of our conditions.
In the case where a ≡ 0, ψ u 0 ∈ L 2 Ω and for more general condition on β, existence results of a solution for parabolic convection diffusion problems 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 |u| r .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 P .We point out that for this class of problems, the regularity assumed on the data f and u 0 , can not expect bounded solutions.
We also point out that we are interested in solutions having finite norms in L 2 0, T; H 1 0 Ω .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 u, existence results.The first ingredient of our proof consists in obtaining certain a priori bounds on the solutions of approximate problems and some suitable L 1 -norm of diffusion terms.A convenient use of Young's inequality will give a uniform estimate of the L 2 0, T; H 1 0 Ω -norms and, therefore, the weak convergence up to a subsequence.We will prove that there exists u such that, up to a subsequence the solution u n of the approximate problems converges to u, all everywhere convergence of gradient of u n to gradient of u, 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 L 1 Ω× 0, T .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 C 0, T ; L 1 Ω , which gives meaning to the initial condition.

Basic Results
Let Ω be a bounded domain in R N , N ≥ 1.We denote by Q T for T > 0 the set Ω× 0, T and by Γ the set ∂Ω× 0, T .
We consider the following nonlinear problem that we denote by P where the unknown function u u x, t is a real function depending on x ∈ Ω and t ∈ 0, T .Λ, A and B are differential operators such that

2.2
Let us consider the following assumptions.
H 1 The real function β is such that The real functions u 0 and f are satisfying 1.7 and 1.6 , respectively.
The operator A x, u is such that where By a weak solution of the problem 1.1 , we mean a function u ∈ L 2 0, T; H 1 0 Ω such that and satisfying for any test function φ in C 1 c Q T the C 1 functions with compact support .
In the sequel we denote by θ a truncation function satisfying θ ∈ C ∞ R , 0 ≤ θ ≤ 1, θ η 1 for |η| ≤ ρ, θ η 0 for |η| 2ρ and |∇θ| ≤ 2/ρ , where ρ is a positive real.By c we denote different constants in R which may vary from line to line.The main result in this paper is the following.

2.8
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 Ω be an open subset of R N with finite Lebesgue measure.Then, for every p such that 1 ≤ p < ∞, one has the following inequality: where ω N is the measure of the unit ball in R N .Furthermore, there exists a constant c c N, p such that, for all u ∈ W 1,P 0 Ω , and the following estimate holds

2.12
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 n.We shall prove some a priori estimates on the solutions of this sequence of problems which serves in the limiting process.

Approximating Problems
We regularize the problem 1.1 by considering the following sequence of problems: where

Let us consider
A n x, u n a x u n |u n | r−1 .

3.3
Next, we consider the truncated function We denote by Ω n a strictly increasing sequence of bounded sets Ω ∩ B n invading Ω.Next we denote From standard results see, e.g., 14 , the following problem: where admits at least one solution u n satisfying

3.8
Then one has the following estimates: Indeed, let us define the following function:

3.12
Let us consider the following sequences:

3.14
Applying Young's inequality, one has the following inequality:

3.15
We now choose q such that q < r, which is possible, since r > 1 and q > 0. We use again Young's inequality twice and we obtain

3.19
Next, we substitute T by t for any t, 0 ≤ t ≤ T in 3.18 , which is possible.We obtain where c is a constant that does not depend on n. then, the approximate problem admits at least one solution which is bounded independently on n in L ∞ 0, T; L r Ω .
Let us now prove that the sequence

3.22
We define, for any k > 0 fixed, the functions h k r defined as follows:

3.23
Let us consider the following sequences:

3.24
We can choose h k u n as test function, and we obtain

3.25
Then, we deduce φ k u 0n x dx.

3.27
Finally, we obtain

3.28
Then, the sequence

ISRN Mathematical Analysis
We require the all everywhere convergence of gradient u n to the gradient of u.Let us consider

3.29
Substituting u in the approximating problem successively with u n and u m , we consider the following function: 3.30 After substraction, for n, m ≥ 4ρ, we obtain the following inequality:

3.31
Let us now consider for n, m sufficiently large the following sequence:

3.32
Then we get

3.34
Using Holder's inequality, we obtain

Limiting Process
We denote by u n the solution of the approximate problems P n on Ω n with initial condition U 0n .To prove the main result, we deal with the limiting process of the approximating problems.First of all, we will prove that there exists u such that, up to a subsequence, u n converges to u, for almost every x, t ∈ Q T .First, we will prove the all everywhere convergence of the gradients of u n to the gradient of u, up to a subsequence, in Q T .Next, we will prove the convergence of the superlinear reaction term and the quadratic gradient term in L 1 Q T .Finally, we will see that u n converges to u in C 0, T ; L 1 Ω , which gives meaning to the initial condition.From 3.38 and up to a subsequence u n , we have ∇u n −→ ∇u a.e. in Q T .

4.1
By consequence, since ∇u n is bounded in L 1 Q T , Vitali's theorem implies that Since one has By a diagonal process, we may select a subsequence, also denoted by {u n }, such that u n −→ u weakly in L 2 0, T; H 1 0 Ω , 4.4