Blow-up solutions for a string equation with nonlinear boundary source and arbitrary-initial-energy

In this paper, we study the following problem y(tt) - y(xx) + y(t) = 0, (x, t) is an element of (0, L) x (0, T) with the initial-boundary conditions {y(0, t) = 0, t is an element of (0, T), y(x)(L, t) = vertical bar y(L, t)vertical bar(p-1)y(L, t) + by(L, t), t is an element of (0, T), y(x, 0) = y(0)(x), y(t)(x, 0) = y(1)(x), x is an element of (0, L), where (0, L) is a bounded open interval in R, p > 1 and b >= 0. We get three sufficient conditions for obtaining the blow-up solutions of the problem. Under some restriction on the parameters in the problem, we conclude that, whether the initial energy is positive or negative, the solution blows up whenever b is large enough. (C) 2012 Published by Elsevier Ltd