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Global nonexistence theorems for quasilinear evolution equations with dissipation. (English) Zbl 0886.35096

The authors show the nonexistence of a global solution of the initial value problem for the abstract evolution equation \[ (P(u_t))_t+ A(u(t))+ Q(t,u_t)= F(u(t)), \] where (1) \(A,F,P\) are Fréchet derivatives of the real-valued functions \({\mathcal A},{\mathcal F},{\mathcal P}\) defined on the Banach spaces \(D,W,V\) with \(W\) continuously imbedded in \(V\) and \({\mathcal A}(0)= {\mathcal F}(0)= {\mathcal P}(0)=0\), (2) \({\mathcal A}(y)\geq 0\) for all \(y\in D\), (3) \({\mathcal P}^*(v)\equiv \langle P(v),v\rangle- {\mathcal P}(v)\geq 0\) for all \(v\in V\). Under suitable assumptions on \({\mathcal A},{\mathcal F},{\mathcal P}\), and \(Q\) it is shown that a solution of the above initial value problem does not exist globally if the initial energy \({\mathcal E}(0)\) is negative or sufficiently negative, where \({\mathcal E}(t)= {\mathcal P}^*(u_t)+ {\mathcal A}(u)- {\mathcal F}(u)\). The results can be applied to the following hyperbolic system \[ (|u_t|^{\ell-2} u_t)_t- a\nabla\cdot (|\nabla u|^{q-2}\nabla u)+ b|u_t|^{m-2}u_t= c|u|^{p-2} u, \] and also to the more general system \[ (|u_t|^{\ell-2} u_t)_t- \nabla\cdot (a(x)|\nabla u|^{q-2}\nabla u)+ Q(t,x,u_t)= F(x,u), \] both with the Dirichlet boundary condition.

MSC:

35L70 Second-order nonlinear hyperbolic equations
58D25 Equations in function spaces; evolution equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
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