On a Higher-order Reaction-diffusion Equation with a Special Medium Void via Potential Well Method

Abstract

Let $d \in \{ 1,2,3,\ldots \}$ and $\Omega \subset \mathbb{R}^d$ be open bounded with Lipschitz boundary. Consider the reaction-diffusion parabolic problem \[ \begin{cases} \frac{u_t}{|x|^4} + \Delta^2 u = k(t) |u|^{p-1}u, &(x,t) \in \Omega \times (0,T), \\ u(x,t) = \frac{\partial u}{\partial \nu}(x,t) = 0, &(x,t) \in \partial \Omega \times (0,T), \\ u(x,0) = u_0(x), &x \in \Omega, \end{cases} \] where $T \gt 0$, $p \in (1,\infty)$, $0 \neq u_0 \in H^2_0(\Omega)$ and $\nu$ is the outward normal vector to $\partial \Omega$. We investigate the existence of a global weak solution to the problem together with the decaying and blow-up properties using the potential well method.