The Cauchy problem for one-dimensional wave equations with a nonlinear dissipative term

Abstract

The Cauchy problem for one-dimensional wave equations with a nonlinear dissipative term is investigated. Under consideration are the problems of uniqueness and existence of local, global and blow-up solutions. The paper’s originality is the coalescence of the two standard methods: a priori estimate of solutions in the class of continuous functions is given by energetic methods; basing on this result a priori estimate in the class of continuously differentiable functions using classical method of characteristics is obtained.