Blow-up of Solutions for a Singular Nonlocal Viscoelastic Equation

Abstract

We study the nonlinear one-dimensional viscoelastic nonlocal problem:  $u_{tt}-\frac{1}{x}(xu_x)_x+ ∫^t_0g(t-s)\frac{1}{x}(xu_x(x,s))_xds=|u|^{p-2}u$, with a nonlocal boundary condition. By the method given in [1, 2], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of blow-up solutions are also given. We improve a nonexistence result in Mesloub and Messaoudi [3].