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On potential wells and applications to semilinear hyperbolic equations and parabolic equations. (English) Zbl 1096.35089

The authors consider both parabolic and hyperbolic partial differential equations. Given the initial boundary value problem for semilinear hyperbolic equations of the form
\[ \begin{aligned} u_{tt}-\triangle u=f(u), \quad & x\in \Omega , \;t>0, \\ u(x,0)=u_0(x), \;u_t(x,0)=u_1(x), \quad & x\in \Omega,\\ u(x,t)=0, \quad &x\in \partial\Omega , \;\;t\geq 0 \end{aligned} \]
and semilinear hyperbolic equations of the form,
\[ \begin{aligned} u_{t}-\triangle u=f(u), \quad & x\in \Omega , \;t>0, \\ u(x,0)=u_0(x), \quad & x\in \Omega ,\\ u(x,t)=0, \quad & x\in \partial\Omega , \;t\geq 0. \end{aligned} \]
Here \(\Omega \) is a bounded domain in \(\text{ I\hskip-2pt R}^n\). The family of potential wells \(W\) is generalized by \[ W=\{u\in H_0^1(\Omega )| \mid (u)>0, J(u)<d\}\cup \{0\}, \] where \(J(u)=(1/2)\| \nabla u\| ^2-\int_{\Omega}F(u)\,dx\), \(F(u)=\int_{0}^{u}f(s)\,ds\), \(I(u)=\| \nabla u\| ^2-\int_{\Omega}uf(u)\,dx\), \(d=\inf{J(u)}\), \(u\in H_0^1(\Omega )\), \(\| \nabla u\| \neq 0\) and \(I(u)=0\). They use and generalize some results of L. E. Payne and D. H. Sattinger [Isr. J. Math. 22 (1975), 273–303 (1976; Zbl 0317.35059)]. A threshold result of global existence and nonexistence of solutions is obtained. The vacuum isolating of the solutions is discussed as well. The global existence of the solutions is proved for the critical initial conditions \(I(u_0)\geq 0\), \(E(0)=d\) (the energy in the initial time) or \(I(u_0)\geq 0\), \(J(u_0)= d\).

MSC:

35L70 Second-order nonlinear hyperbolic equations
35K55 Nonlinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations

Citations:

Zbl 0317.35059
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References:

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