A note on global strong solutions of semilinear wave equations (Q1814216)

The author studies the following problem: \[ L(u)+F(u)=0\quad \hbox {in } [0,\infty[\times\Omega \leqno (*) \] \[ u(0,x)=f(x), \quad\partial_ tu(0,x)=g(x)\quad \hbox {in }\Omega \] \[ Bu(t,x)=0 \quad \hbox {on } [0,\infty[\times\partial\Omega, \] where \(L\) is a linear hyperbolic partial differential operator of second order with coefficients depending on \(t\in[0,\infty[\), \(x=(x_ 1,\ldots,x_ n)\in\mathbb{R}^ n\) and \(\Omega\) is a bounded or unbounded domain of \(\mathbb{R}^ n\) with a smooth boundary \(\partial\Omega\), or the whole \(\mathbb{R}^ n\). He assumes that \(3\leq n\leq 6\), the nonlinear term \(F(u)\) is a regular function satisfying \(| F(u)|\leq\hbox{const}(| u|+| u|^{n/(n-2)})\) as \(| u|\to\infty\) and, in case \(\Omega\neq\mathbb{R}^ n\), \(B\) is a suitable boundary linear operator. The author obtains the existence and uniqueness of classical solutions of the problem (*). For that, first, by applying the contracting mapping principle, he gets a unique classical solution of (*) in some finite interval of \([0,\infty[\). Then by appropriate a priori estimates of solutions, he obtains a solution defined in the whole \([0,\infty[\). In this approach, he recourses to some known results on existence, uniqueness and energy estimates of solutions for linear hyperbolic partial differential equations.