On space-time means and strong global solutions of nonlinear hyperbolic equations
The author studies the following nonlinear problem: \[ (*)\quad \partial^ 2_ tu-\Delta u+b(x,D)u+f(u)=0,\quad (t,x)\in {\mathbb{R}}_+\times {\mathbb{R}}^ n, \] \[ u(0,x)=\phi_ 0(x),\quad (\partial /\partial t)u(0,x)=\phi_ 1(x),\quad x\in {\mathbb{R}}^ n, \] where b(x,D) is any first order differential operator on \({\mathbb{R}}^ n\) with bounded smooth coefficients and f is a real function defined in \({\mathbb{R}}\). Under the hypothesis \(\phi_ 0\in L^ 2_ 2({\mathbb{R}}^ n)\), \(\phi_ 1\in L^ 1_ 2({\mathbb{R}}^ n)\) and \(f\in C^ 2\), \(f(0)=0\), \(F(\sigma)=\int^{\sigma}_{0}f(u)du\geq 0\), \(\forall \sigma \in {\mathbb{R}}\), \[ | f'(u)| \leq C(1+| u|)^{\rho -1},\quad 1<\rho <(n+2)/(n-3),\quad n\geq 3, \] he shows that there exists a unique solution u of problem (*) belonging to \(L_{\infty}^{loc}({\mathbb{R}};L^ 2_ 2({\mathbb{R}}^ n))\). For that, first he obtains the same result for a nonlinear Klein-Gordon equation with nonlinear term f(u) and data \(\phi_ 0\), \(\phi_ 1\). Then he applies this result to the general case (*). In his approach he uses properties of the solution of the corresponding linear Klein-Gordon equation and appropriate estimates for the nonlinear term.
scientific article; zbMATH DE number 4157086