Abstract
We consider the semilinear wave equation with time-dependent damping
partial derivative(tt)u - Delta u + mu(1 + t)(-beta)partial derivative(t)u = vertical bar u vertical bar(p), (t, x) is an element of (0, infinity) x D-c,
where D-c = R-N\D, D is the closed unit ball in R-N, N >= 2, mu > 0, p > 1 and -1 < beta < 1. The considered equation is investigated under the boundary conditions:
u(t, x) (or partial derivative u/partial derivative n(+) (t, x)) = b(t) f(x) on (0, infinity) x partial derivative D,
where n(+) is the outward (relative to D-c) unit normal on partial derivative D. General blow-up results are established for the considered problems. Moreover, for a certain class of functions b, the critical exponent in the sense of Fujita is obtained.