Some remarks on the wave equations with nonlinear damping and source terms. (English) Zbl 0866.35071

The author studies the following mixed problem \[ u_{tt}- \Delta u+ \delta|u_t |^{m-1} u_t= |u|^{p-1}u, \quad x\in\Omega,\;t\geq 0, \tag{1} \]
\[ u(0,x)= u_0(x),\;u_t(0,x) =u_1(x),\;x\in\Omega, \quad u(t,x)= 0,\;x\in\partial \Omega,\;t\geq 0. \] Here \(\Omega \subset\mathbb{R}^N\) is a bounded domain with smooth boundary \(\partial\Omega\), \(p>1\), \(m\geq 1\), \(\delta>0\) and \(\Delta\) is the Laplacian in \(\mathbb{R}^N\). The maximal time interval of existence of the solution is such that (1) admits a solution \(u(t,\cdot) \in C([0,T_m); H^1_0 (\Omega)) \cap C^1([0,T_m); L^2(\Omega))\) with \(u_t\in L^\infty ((0,T)\times \Omega)\) for any \(0<T< T_m\) and if \(T_m<\infty\), then \[ \lim_{t\nearrow T_m} \bigl|\nabla u(t,\circ) \bigr|_2+ \bigl|u_t(t,\circ) \bigr|_2= \infty. \] To study the existence of a stable set for this problem, the author introduces the notion of potential depth as follows: \[ J(u)= {1\over 2} |\nabla u |^2_2- {1 \over p+1} |u |^{p+1}_{p+1}, \] \(E(u,v)= {1\over 2} |v|^2_2 +J(u)\), \(I(u)= |\nabla u |^2_2-|u |^{p+1}_{p+1}\), \(d\equiv \inf\{\sup_{\lambda \geq 0} J(\lambda u)\); \(u\in H^1_0 (\Omega),\;u\neq 0\}\). It is well known that the potential depth \(d\) is positive. The stable set is then defined by \(W^*= \{u\in H^1_0(\Omega)\); \(J(u)<d\), \(I(u)>0\} \cup \{0\}\). The first result is the following.
Theorem 1. Suppose \(\delta>0\), \(m\geq 1\) and either \(p>1\) \((N=1,2)\) or \(1<p \leq N/(N-2)\). Let \(u(t,x)\) be a local solution of (1) and \([0,T_m)\) the maximal time-interval of existence of the solution. If there exists a real number \(t_0\in [0,T_m)\) such that \(u(t_0,\circ)\in W^*\) and \(E(u(t_0,\circ), u_t(t_0,\circ)<d\), then \(T_m= \infty\).
Under the additional assumptions that either \(m\geq 1\) \((N=1,2)\) or \(1\leq m\leq {N+2 \over N-2}\) \((N\geq 3)\) the author shows that the following two conditions are equivalent:
i) There exists a real number \(t_0\in[0,T_m)\) such that \(u(t_0,\circ)\in W^*\) and \(E(u(t_0,\circ)\), \(u_t [t_0,\circ))<d\),
ii) \(T_m = \infty\) and \(\lim_{t\to\infty} |\nabla u(t,\circ)|_2=\lim_{t\to\infty} |u_t(t,\circ) |_2=0\).
Reviewer: V.Georgiev (Sofia)


35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
Full Text: DOI


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