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

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.