Existence of global solutions for a reaction-diffusion equation with singular coefficients and conical degeneration

Abstract

Let $d\in \{4,5,6,\dots \}$ and let $X$ be a closed compact manifold of dimension $d-1$. Let $\mathbb{B}=[0,1)\times X$ be a conical domain and $\mathring{\mathbb{B}}$ its interior. Consider the reaction-diffusion parabolic problem

where $T>0$, $p\in (1,\infty )$, $u_0$ belongs to the cone Sobolev space ${\mathcal{H}}_{2,0}^{1,d/2}(\mathbb{B})$ and ${\Delta }_{\mathbb{B}}$ is the Fuchsian Laplacian. We investigate the existence of a global solution to (P) together with the decaying and blow-up properties using the potential well method.