On a nonlinear heat equation associated with Dirichlet -- Robin conditions

TL;DR

This paper investigates a nonlinear heat equation with Dirichlet-Robin boundary conditions, establishing existence, uniqueness, boundedness, asymptotic behavior of solutions, and providing numerical results.

Contribution

It introduces new analytical results for nonlinear heat equations with mixed boundary conditions, including existence, uniqueness, and long-term behavior analysis.

Findings

Solutions remain bounded if initial conditions are bounded

Asymptotic behavior of solutions characterized

Numerical simulations support theoretical results

Abstract

This paper is devoted to the study of a nonlinear heat equation associated with Dirichlet-Robin conditions. At first, we use the Faedo -- Galerkin and the compactness method to prove existence and uniqueness results. Next, we consider the properties of solutions. We obtain that if the initial condition is bounded then so is the solution and we also get asymptotic behavior of solutions as. Finally, we give numerical results