Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation

Abstract

.We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $\ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, & (t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, & (t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es& \ea$ where $<\le \infty$, $\Omega$ is a bounded regular open subset of $\mathbb{R}^n$, $n\ge 1$, $b,c>0$, $p>2$, $m>1$. We prove a global nonexistence theorem for positive initial value of the energy when $1<m<p,\quad 2, <p\le \frac{2n}{n-2}.$ We also give applications concerning the classical equations of linear elasticity, the damped clamped plate equation and evolution systems involving the q‐Laplacian operator, $q>1$.