Lower bound of the lifespan of solutions to semilinear wave equations in an exterior domain (Q2842364)

In this article is considered the Cauchy-Dirichlet problem for semilinear wave equations NEWLINE\[NEWLINEu_{tt}-\Delta_xu=F(\partial u),\quad (t, x)\in (0, T)\times \Omega, \tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(t, x)=0,\quad (t, x)\in (0, T)\times\partial\Omega, \tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0, x)=\epsilon f_0(x), (\partial_t u)(0, x)=\epsilon f_1(x), \quad x\in \Omega, \tag{3}NEWLINE\]NEWLINE where \(\Delta_x=\sum_{i=1}^3 \partial_{x_i}^2\), \(\partial u=(\partial_t u, \nabla_x u)\), \(\epsilon\) is a small and positive parameter, \(\Omega=\mathbb R^3\backslash O\), \(O\) is a non-trapping obstacle, \(F(\partial u)=\sum_{a, b=0}^3 g^{a, b}(\partial_a u)(\partial_b u)\) with real constants \(g^{a, b}\), \(\partial_0=\partial/\partial_t\), \(\partial_i=\partial/\partial_{x_i}\), \(i=1, 2, 3\), \((f_0, f_1)\in ({\mathcal{C}}_0^{\infty}(\Omega))^2\).NEWLINENEWLINEThe authors investigate the behaviour of the lifespan \(T_{\epsilon}\) of the solution \(u\) of the problem (1)--(3). In the article the lifespan \(T_{\epsilon}\) is defined by the supremum of all \(T>0\) such that the mixed problem (1)--(3) admits a unique smooth solution \(u\) in \([0, T)\times \Omega\). The authors give conditions for \(F\) and \((f_0, f_1)\) so that \(\liminf_{\epsilon\longrightarrow +0}\epsilon \log T_{\epsilon}\geq \tau_{*}\), where NEWLINE\[NEWLINE\tau_{*}=(\sup\{-2^{-1}G(\theta){\mathcal{F}}_+(s, \theta):s\in\mathbb R, \theta\in S^2\})^{-1},NEWLINE\]NEWLINE \(G(\theta)=\sum_{a, b=0}^3 g^{a, b}\theta_a\theta_b\), \({\mathcal{F}}_+(s, \theta)=\lim_{t\longrightarrow\infty}(-t)(\partial_t u_0)(t, (s+t)\theta)\), \(\theta=(\theta_1, \theta_2, \theta_3)\in S^2\), \(\theta_0=-1\).NEWLINENEWLINEIn the article are given some special cases for \(F\) and \((f_0, f_1)\) when \(O=\{x\in \mathbb R^3:|x|<1\}\) so that \(\limsup_{\epsilon\longrightarrow +0}\epsilon \log T_{\epsilon}\leq \tau_{*}\).NEWLINENEWLINEThe authors extend their result for some quasilinear wave equations with the Dirichlet condition (2) and initial data (3).