The Nature Of The Two Types Of Nonlinear Partial Differential Equations

With the development of science and technology, all kinds of nonlinear prob-lems have aroused people's wide attention. Nonlinear partial differential equa-tions origin from applied mathematics, physics, control theory and other applied sciences, they are the most active research topics in the field of nonlinear science. The initial-boundary value problems of nonlinear viscoelastic wave equations and nonlinear integro-differential equations are hot topics these years, and they are important research fields of the partial differential equations.This dissertation is devided into two chapters.In the first chapter, we study the decay for the following inital-boundary problem of nonlinear system of viscoelastic wave system, whereΩis a bounded domain in Rn(n≥1)with a smooth boundary ()Ω. The functions u0,v0,u1, and v1are given initial functions. The relaxation functions h1,h2 and the nonlinearities k1(u,v), k2(u, v) satisfy the following conditions:(H1) h1, h2:R+â†'R+are nonincreasing differentiable functions satisfying(H2) There exist two positive differentiable functionsξ1 andξ2 such that (H3) There exist a nonnegative function F(u, v) and a constant d>0 such that whereRemark:There exist some functions satisfying (H3), for example whereThis chapter includes three sections. In the first section, we give some im-portant results which can be used during proof. In the second section, in order to prove the global solutions decay, we give some preliminary knowledge. In the third section, we prove the global solutions decay for the nonlinear system of viscoelastic wave equations.In the second chapter, we study the backwards uniqueness in time of solu-tions to a class of nonlinear integro-differential system (2.1.1) with Neumann or Dirichlet boundary condition, with p≥2,Ωbe a bounded domain of Rn with a sufficiently smooth boundaryΓ, andΩT:=Ω(0, T]. The vector v=(v1, v2,…, vn) is the unit exterior normal, where This chapter contains two sections. In the first section, we study the non-linear integro-differential system with Neumann boundary condition, and give reasonable physical interpretation for our conclusions. In the second section, us-ing the methods similar to those in the first section, we study this system with Dirichlet boundary condition.