Some existence results for a class of parabolic equations with nonlinear boundary conditions

Anass Lamaizi; Mahmoud El Ahmadi; Mohammed Barghouthe; Omar  Darhouche

doi:10.24193/subbmath.2025.4.08

Authors

Anass Lamaizi Department of Mathematics, Faculty of Sciences and Technology, Abdelmalek Essaadi University, Tetouan, Morocco https://orcid.org/0000-0003-4469-6460
Mahmoud El Ahmadi Department of Mathematics, Faculty of Sciences and Technology, Sidi Mohamed Ben Abdellah University, Fez, Morocco https://orcid.org/0000-0002-6493-3219
Mohammed Barghouthe Department of Mathematics, Faculty of Sciences, Mohammed I University, Oujda, Morocco https://orcid.org/0000-0002-5714-0157
Omar Darhouche Department of Mathematics, Faculty of Sciences and Technology, Abdelmalek Essaadi University, Tetouan, Morocco https://orcid.org/0000-0003-2604-1700

DOI:

https://doi.org/10.24193/subbmath.2025.4.08

Keywords:

parabolic problem, global existence, blow up

Abstract

In this paper, we are interested to study the weak solutions for the following nonlinear parabolic problem: \[ \begin{cases} u_t - \Delta_p u + \vert u \vert^{p-2} u = 0 \quad \text{ in } ~ \Omega ,~ t>0 , \\ \vert \nabla u \vert^{p-2} \frac{\partial u}{\partial \nu }= g(u) \quad \quad \quad ~~ \text{ on }~ \partial \Omega ,~ t>0 , \\ u(x;0)=u_0 (x) \quad \quad \quad \quad ~~~~ \text{ in } ~ \Omega . \end{cases}
\] Using the Galerkin approximation and a family of potential wells, we establish the existence of global weak solution under appropriate conditions. Additionally, we provide a result on the blow-up and asymptotic behavior of certain solutions with positive initial energy.

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