Existence and blow-up of solutions for a class of semilinear pseudo-parabolic equations with cone degenerate viscoelastic term

Abstract

In this paper, we consider the semilinear pseudo-parabolic equation with cone degenerate viscoelastic term

$$\begin{aligned} u_t+\Delta _{\mathbb B}^{2} u_t+\Delta _{\mathbb B}^{2}u-\int _0^t g(t-s)\Delta _{\mathbb B}^{2}u(s)ds=f(u),\ \text{ in } \text{ int }\mathbb B\times (0,T), \end{aligned}$$

with initial and boundary conditions, where \(f(u)=|u|^{p-2}u-\frac{1}{|\mathbb B|}\displaystyle \int _{\mathbb B}|u|^{p-2}u\frac{dx_1}{x_1}dx'\). We construct several conditions for initial data which leads to global existence of the solutions or the solutions blowing up in finite time. Moreover, the asymptotic behavior and the bounds of blow-up time for the solutions are given.