Global nonexistence theorems for a class of evolution equations with dissipation

The paper proves a new global non-existence result for a large class of nonlineary damped evolution equations. In particular it considers the case of positive initial energy.

We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the typeu(tt) - Delta u + b/u(t)/(m-2)u(t) = c/u/(p-2)u (t, x) epsilon [0, T) x Omega,u(t, x) = 0, (t, x) epsilon [0, T) x partial derivative Omega,u(0,.) = u(0) epsilon H-0(1) (Omega), u(t)(0,.) = v(0) epsilon L-2 (Omega),where 0 < T less than or equal to infinity, Omega is a bounded regular open subset of R-n, n greater than or equal to 1, b, c > 0, p > 2, m > 1. We prove a global nonexistence theorem for positive initial value of the energy when1 < m < p, 2 < p less than or equal to 2n/n-2We also give applications concerning the classical equations of linear elasticity, the damped: clamped plate equation and evolution systems involving the q-Laplacian operator, q > 1.