Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms

Abstract

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for a class of quasi-linear wave equations with nonlinear damping and source terms. It proves that when α ⩾ max { m , p } , where m + 1 , α + 1 and p + 1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, under rather mild conditions on initial data, the Cauchy problem admits a global weak solution. Especially in the case of space dimension N = 1 , the weak solutions are regularized and so generalized and classical solution both prove to be unique. On the other hand, if 0 ⩽ α < 1 , and the initial energy is negative, then under certain opposite conditions, any weak solution of the Cauchy problem blows up in finite time. And an example is shown.