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Explosion for some semilinear wave equations. (English) Zbl 0679.35068
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.
Reviewer: V.Georgiev

MSC:
35L70 Second-order nonlinear hyperbolic equations
35G25 Initial value problems for nonlinear higher-order PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35L05 Wave equation
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