TY - JOUR T1 - Blow-up of Solutions for a Singular Nonlocal Viscoelastic Equation JO - Journal of Partial Differential Equations VL - 2 SP - 140 EP - 149 PY - 2011 DA - 2011/05 SN - 24 DO - http://doi.org/10.4208/jpde.v24.n2.3 UR - https://global-sci.org/intro/article_detail/jpde/5202.html KW - Blow-up KW - life span KW - viscoelastic KW - nonlocal problem AB -

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].