International Journal of Nonlinear Analysis and Applications

Quasilinear parabolic problems in the Lebsgue-Sobolev space with variable exponent and $L^1$ data

Document Type : Review articles

Authors

1 University 20th August 1955, Skikda, Algeria

2 Laboratory of Applied Mathematics and History and Didactics of Maths "LAMAHIS", Algeria

Abstract

In this work, we study the existence of an initial boundary problem of a quasilinear parabolic problem with variable exponent  and  $ L ^{1} $-data of the type
\begin{equation*}
\left\{
\begin{array}{ll}
(b(u))_{t}-\text{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)+\lambda
\left\vert u\right\vert ^{p(x)-2}u=f(x,t,u)
 \text{ } &
\text{in}\hspace{0.5cm}Q=\Omega \times ]0,T[, \\
u=0 & \text{on}\hspace{0.5cm}\Sigma =\partial \Omega \times ]0,T[, \\
b(u)(t=0)=b(u_{0}) & \text{in}\hspace{0.5cm}\Omega , 
\end{array}
\right.
\end{equation*}
where $ \lambda>0$ and $ T $ is positive constant. The main contribution of our work is to prove the existence of a renormalized solution. The functional setting involves Lebesgue– Sobolev spaces with variable exponents.

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 10
October 2024
Pages 117-130
  • Receive Date: 01 May 2023
  • Revise Date: 16 August 2023
  • Accept Date: 18 August 2023