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


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
Full Text: DOI Link


[1] Bachelot, A., Equipartition de l’énergie pour les systèmes hyperbolique et formes compatibles, C. R. Acad. Sci. Paris Sér. I Math., 301, No. 11, 573-576 (1985) · Zbl 0601.35067
[2] Hanouzet, B.; Joly, J. L., Explosion pour les problèmes hyperboliques semi-linéaire avec second membre non compatible, C. R. Acad. Sci. Paris Sér. I Math., 301, No. 11, 581-584 (1985) · Zbl 0601.35073
[3] Hanouzet, B.; Joly, J. L., Applications bilinéaires compatibles avec un système hyperbolique, C. R. Acad. Sci. Paris Sér. I Math., 301, No. 10, 491-494 (1985) · Zbl 0601.35066
[4] Klainerman, S., Global existence for nonlinear wave equations, Comm. Pure Appl. Math., 33, 43-101 (1980) · Zbl 0405.35056
[5] Rauch, J.; Reed, M., Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: Creation and propagation, Comm. Math. Phys., 81, 203-227 (1981) · Zbl 0468.35064
[6] Rauch, J.; Reed, M., Discontinuous progressing waves for semilinear systems, Comm. Partial Differential Equations, 10, 1033-1076 (1985) · Zbl 0598.35069
[7] L. Tartar, Preprint Mathematic Research Center, University of Wisconsin.; L. Tartar, Preprint Mathematic Research Center, University of Wisconsin.
[8] Balabane, M., Ondes progressive et résultats d’explosion pour des systèmes non linéaires du premier ordre, C. R. Acad. Sci. Paris Sér. I Math., 302, No. 6 (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.