Asymptotic Properties Of Singular Solutions In Parabolic Equation With Variable Exponents And Damping Source

In this paper,we study a class of nonlinear damping and anisotropic growth condition variable index of parabolic partial differential equation,the nonlinear with variable index from local and non-local internal source and absorb the source in the form of the product.According to the basic theory of parabolic equation,we construct the energy function and control function to acquire the asymptotic properties of the blow-up solutions,including blow-up criteria of solutions,global existence,blow-up set,blow-up time estimate and blow-up rate estimate.The relationship between the asymptotic property and the coefficient of reaction,the coefficient of the damping coefficient and the number of variable exponents are given.This article is divided into seven chapters:The first and second chapters of the article firstly introduce the background,significance and current research status of the present study,then give the questions and necessary knowledge reserve.In the third chapter,the energy function method and the variable exponentia modified Kaplan method are used to solve the blasting conditions,respectively.In chapter 4,by constructing the proper solution,the whole existence of understanding is proved by the comparison principle.In chapter fifth,six,seven,by constructing auxiliary function,asymptotic properties of the related results are obtained,and the main results are the upper and lower bounds of the blasting time the upper and lower bounds of the estimates,blasting rate estimation and the existence of the blasting set.It is found that the variable exponent and coefficient play an important role in the Fujita blow-up phenomena of the parabolic equation with variable exponent.