The Asymptotic Behavior Of Viscoelastic Wave Equation

The nonlinear partial differential equation(PDE)is an important branch of the modern mathematics.In the field of physics,chemistry,biology and other fields,the nonlinear partial differential equation can be used to describe the theoretical and practical applications.In the paper,we study the asymptotic behavior of solutions for the stochastic viscoelastic wave equation derived from physics or cybernetics to describe the matter particles position of elastic materials and viscous materials mixed model.The main results are as follows:In chapter 3,we discuss an initial boundary value problem of stochastic viscoelastic wave equatiol driven by multiplicative noise involving the nonlinear damping term|ut|q-2 ut and a source term of the type |u|p-2 u.We first establish the local existence and uniqueness of solution by the iterative technique truncation function method.Moreover,we also show that the solution is global for q?p.Lastly,by modifying the energy functional,we give sufficient conditions such that the local solution of the stochastic equations will blow up with positive probability or explode in energy sense for q<p.At the same time,we get the noise term to delay the explosive of the solution.In chapter 4,we discuss a class of second order stochastic evolution equations with memory is considered.First we give the definition of the equation solution under the appropriate relationship between the assumptions of the functions g,f and the coefficients of the source terms.Then we use the appropriate energy inequalities and stopping time techniques to derive the maximum local mild solution.Lastly,by Tartar's energy method,we prove that for any solution to the equation with probability one,the speed of propagation is with velocity less or equal 1.