Life-span of classical solutions to one dimensional initial–boundary value problems for general quasilinear wave equations with Robin boundary conditions

Abstract

This paper is devoted to studying the initial–boundary value problem for one dimensional general quasilinear wave equations utt−uxx=b(u,Du)uxx+2a0(u,Du)utx+F(u,Du) with Robin boundary conditions on an exterior domain. We obtain the sharp lower bound of the life-span of classical solutions to the initial–boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations. Our result in the general case and the special case is shorter than that of the initial–boundary value problem for one dimensional general quasilinear wave equations with Dirichlet boundary conditions. The results in this paper are not the trivial generalization of that in the case of Dirichlet boundary conditions. The lower bound estimates of life span of classical solutions to initial–boundary value problems are consistent with the actual physical meaning. The physical phenomenon also explains our results.