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 - <p>We study the nonlinear one-dimensional viscoelastic nonlocal problem:&nbsp; $u_{tt}-\frac{1}{x}(xu_x)_x+ ∫^t_0g(t-s)\frac{1}{x}(xu_x(x,s))_xds=|u|^{p-2}u$, 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].</p>