Quasilinear class of noncoercive parabolic problems with Hardy potential and L 1-data

Taghi Ahmedatt: Department of Mathematics and Computer Science, Laboratory G3A, Faculty of Sciences and Technology, University of Nouakchott, P. Box 880 Nouakchott, Mauritania, e-mail: taghi-med@hotmail.fr Youssef Hajji: Department of Mathematics, Faculty of Sciences, University Abdelmalek Essaadi, BP 2121, Tetouan, Morocco, e-mail: youssef.hajji@etu.uae.ac.ma  * Corresponding author: Hassane Hjiaj, Department of Mathematics, Faculty of Sciences, University Abdelmalek Essaadi, BP 2121, Tetouan, Morocco, e-mail: hjiajhassane@yahoo.fr Nonautonomous Dynamical Systems 2023; 10: 20220168

They proved the existence of positive solutions in the absorption case u 2 | | + ∇ for all λ 0 > and f L Ω 1 ( ) ∈ . Furthermore, they showed the nonexistence of solution in the diffusion case u 2 | | − ∇ and λ 0 > (even in a very weak sense) (see also [1] and [2]). These problems are related to the following classical Hardy inequality: is optimal and is not attained; we refer to [7] for more details about Hardy inequality. In [20], Porzio studied the quasilinear elliptic problem < + (see also [10,11,14,17,21,24]).
In [18], Porretta and Segura de León investigated the existence results of the problem a x u u g x u u f x u div , , , , in Ω, 0 o n Ω , where a x s ξ , , ( )verifying a degenerate coercivity condition, and g x s ξ , , ( )is assumed to satisfy only some growth conditions. They proved the existence of solutions using rearrangement techniques, see also [22] and [12], for the unilateral case we refer the reader to [13]. In and proved the existence of solutions and some regularity results in the case of f in L Ω m ( ), with m 1 ≥ (see also [5]).
For T 0, > we denote by Q T the cylinder and by Σ T the lateral surface T Ω 0, . ( ) ∂ × Baras and Goldstein [8]  the authors proved the nonexistence of a local solution for any u 0. 0 > Azorero and Alonso [7] studied the behavior of the nonlinear critical p-heat equation They showed the existence of a solution for λ λ .
Moreover, they proved the nonexistence of a local solution for any f 0 ( ) ⋅ > for λ λ N > , we refer the reader also to [19]. In this article, we study the existence of solutions for the problem associated with quasilinear parabolic equations involving a Leray-Lions-type operator with lower order terms and the so-called Hardy potential , and we consider the assumptions , and max 1 , Note that this article can be seen as a continuation of [7] and [8] for the nonlinear and noncoercive cases by adding the lower order term u u s 1 | | − , which guarantees the existence of entropy solutions for any λ 0. ≥ This article is organized as follows: in Section 2, we present some assumptions on a x t s ξ , , , ( ) for which our noncoercive parabolic problem (1.8) has at least one entropy solution. Section 3 contains some important lemmas useful to prove our main result. Section 4 will be devoted to the proof of the existence of entropy solutions for our quasilinear noncoercive parabolic problem (1.8).

Essential assumption
Let Ω be a bounded open subset of N 2 N ( ) ≥ containing the origin, T 0 > , and p .
for almost every x t , ( ) in Q T ), which satisfies the following conditions: for a.e.
x Ω ∈ and all s ξ ,  , and we assume that 3 Some technical lemmas In view of Lemma 3.1, we obtain Moreover, in view of [9], we have Now, let μ 0, ≥ and we introduce the time mollification u μ of a function u L T W 0, ; Ω such that u verifying the inequality Remark 4.1. In view of Young's inequality, we have Step 1: Approximate problems in Ω.

4.1.2
Step 2: A priori estimates Lemma 4.3. Under the assumptions of Theorem 4.2, there exists a constant C 0 > , not depending on n, such that the following estimates hold true: Proof of the Lemma 4.3. Let θ 1, > by taking ψ u u 1 s i g n n u n Using (2.4) and the fact that ψ u We have Concerning the first term on the right-hand side of (4.9), we set ω s s θ s θ θ s s θ The same result can be obtained directly from (4.11) in the case of θ 2 > . In the case of θ 2, = we have By combining (4.11) and (4.12), we deduce that for any θ In view of (4.14), we obtain that for all ε 0, > there exists k ε 0 0 ( ) > such that u k ε u k ε k k ε meas 3 and meas 3 .
In view of (4.6), the sequence T u It follows that u n n ( ) is a Cauchy sequence in measure, then there exists a subsequence still denoted u n n ( ) such that and in view of the Lebesgue dominated convergence theorem,

( ( ) )+ ( )
as a test function in (4.3), we obtain  For the first term on the right-hand side of (4.20), we have Concerning the last three terms on the right-hand side of (4.20), by using Young's inequality, we have .
By combining (4.20)-(4.21), we deduce that with C 11 being a constant that does not depend on n, then the sequence
It follows that In view of Young's inequality, we have On the other hand, for any measurable subset E Q T ⊂ , we have and there exists β η 0 ( ) > such that 4.1. 6 Step 6 Convergence of the gradient In the sequel, we denote by ε n j ( ), j 1, 2, = … some various functions of real numbers, which converge to 0 as n tends to infinity. Similarly, we define ε h j ( ), ε n h , j ( ) and ε n μ h , , j ( ).    For the first term on the right-hand side of (4.34), we have