Explosion for some semilinear wave equations
The author studies the explosion of the solution to the semilinear wave equation with quadratic nonlinearity. First of all, he treats the one- dimensional case and reduces the semilinear wave equation as well as the Carleman model of transport theory to the following nonlinear (2\(\times 2)\) system \[ (1)\quad \partial_{\pm}u_{\pm}=f_{\pm}(u_+,u_- ),\quad \partial_{\pm}=\partial_ t\pm \partial_ x,\quad f_{\pm}=A_{\pm}u^ 2_{\pm}+B_{\pm}u_+u_-+C_{\pm}u^ 2_{\pm}, \] where \(A_{\pm}\), \(B_{\pm}\), \(C_{\pm}\) are constants. The first result asserts that, when \(A_+\neq 0\) or \(C_-\neq 0\), one can find initial data \(u^ 0_{\pm}\in L^{\infty}({\mathbb{R}})\), piecewise \(C^{\infty}\) and of compact support such that the Cauchy problem for (1) with initial data \(u_{\pm}(0,x)=\epsilon u^ 0_{\pm}(x)\) does not have a global solution for any \(\epsilon\neq 0\). The proof is based on comparison of \(L^{\infty}\) norm of \(u_+\), \(u_-\) with the non- negative solution of a suitable ordinary differential equation. The case of the nonlinear wave equation in higher dimensional space is considered too.
scientific article; zbMATH DE number 4113165