Global Existence, Uniform Decay And Blow-up Of Solutions For Nonlinear Viscoelastic Wave Equations Of Kirchhoff Type

This work concerns with nonlinear viscoelastic wave equations of Kirchhoff type and related results concerning local existence, global existence, asymptotic behavior and blow-up of solutions have been established.In the first chapter we review some development of the related problems and summarize the main work of the present dissertation. Moreover, we introduce the spaces and notations.In Chapter2, we consider the nonlinear viscoelastic wave equations of Kirchhoff type with Dirchlet boundary conditions. Under some suitable assumption on g and the initial data, we establish a global nonexistence result for certain solutions with arbitrarily high energy.The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with strong damping is considered in Chapter3. Firstly, we prove the local existence of solutions by using the Faedo-Galerkin approximation and the contraction mapping principle. Then, we establish two blow-up results:one is for certain solutions with non-positive initial energy as well as positive initial energy by exploiting the convexity technique, the other is for certain solutions with arbitrarily positive initial energy based on the method of Li and Tsai [26]. Finally, we give a general decay result of global solutions by the perturbed energy method under a weaker assumption on the relaxation functions.