Blow-up And Asymptotic Behavior Of Solutions Of Two Kinds Of Delay Partial Differential Equations

A large number of models in many applied disciplines,such as biology,physics,economics,chemistry,ecology,and finance,are generally applied to partial differential equations.One of the branches that are particularly active and growing rapidly in mathematics is stochastic partial differential equations.In this paper,we study the blow-up and asymptotic behavior of solutions of two kinds of delay partial differential equations.The main contents are as follows:Firstly,this paper studies a class of stochastic reaction-diffusion equations with time-delay terms,considering the blasting of the solution driven by additional noise and multiplicative noise.Using the comparison principle and Jensen's inequality,the exponential competition between time delay terms,diffusion terms,and random terms is given,and the result of noise terms delaying the blasting of solutions is obtained.The specific results are as follows:(1)Under the driving of additional noise,obtain the solution to blast at a finite time T',and give an upper bound estimate of the blow-up time;(2)Driven by multiplicative noise ?(u)=ua(x,t),if 1/2<a<1,the solution is exploded under the condition of the diffusion term index p>1 and the time-delay term index q?0;If a>1,the solution is exploded under the condition of p>2a-1.Secondly,the stability of the solution of the variable coefficient viscoelastic wave equation with time-delay term on the bounded region with smooth boundary is discussed.Based on the Riemannian geometry method and some estimates of the viscoelastic wave equation,for the case of|?2|<?1 and |?2|=?1,the appropriate Lyapunov functionals are constructed respectively,and the asymptotic results of the exponentially decreasing energy of the solution of the viscoelastic wave equation with variable coefficients are obtained when t is infinite,and the internal feedback of the linear dissipative delay with time delay is obtained.Finally,the invariant measure of a class of stochastic viscoelastic wave equations driven by a non-Gaussler process is discussed.The existence and uniqueness of the local weak solution are proved by the post-push method,and then under the appropriate conditions,given the existence and uniqueness of the invariant measure of the transferred semigroup generated by the mild solution.