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Archive for Rational Mechanics and Analysis

, Volume 149, Issue 2, pp 155–182 | Cite as

Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation

  • Enzo Vitillaro
Article

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, \(q>1\).

Keywords

Wave Equation Classical Equation Evolution Equation Open Subset Evolution System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Enzo Vitillaro
    • 1
  1. 1.Dipartimento di Matematica ed Informatica, Università di Perugia, Via Vanvitelli, 1, 06123 Perugia, Italy, e-mail: enzo@unipg.itIT

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