The Global existence and blow-up of solutions for a class of Steklov parabolic problems

Abstract

In this paper, we study weak solutions to the following Steklov parabolic problem: $$ \begin{cases} u_t - \Delta_p u + \vert u \vert^{p-2} u = 0 \quad \quad \text{ in } ~ \Omega ,~ t>0 , \\ \vert \nabla u \vert^{p-2} \frac{\partial u}{\partial \nu }= \lambda \vert u \vert ^{q} u \quad \quad \quad ~ \text{ on }~ \partial \Omega ,~ t>0 , \\ u(x;0)=u_0 (x) \quad \quad \quad \quad \quad ~~ \text{ in } ~ \Omega . \end{cases} $$ where $\Omega \subset \mathbb{R}^{n}$ is an open bounded domain for $ n \geq 2$ with smooth boundary $ \partial \Omega$, $\lambda > 0 $. Here, $u_t $ denote the partial derivative with respect to the time variable $t$ and $ \nabla u$ denotes the one with respect to the space variable $x$. We prove theorems of existence of weak solutions, via Galerkin approximation. Moreover, we show the existence of solutions which blow up in a finite time.