Asymptotic Properties Of Singular Solutions In A Parabolic Equation With P(x,t)-Laplacian

This thesis deals with the homogeneous Dirichlet problem of a nonlinear diffusion equation involving anisotropic variable and convection.Firstly,we construct a space with anisotropic Orlicz-Sobolev,prove its completeness,separability and dense subset,and give the completeness of the corresponding conjugate space.Then,the existence and uniqueness of weak solutions are proved,and the regularity of weak solutions is improved under certain conditions.Secondly,we overcome the influence of the variable index,construct different forms of energy functions,and combine the convexity method to give the blasting criterion of the solution.In addition,we use the comparison principle of ordinary differential equations to construct auxiliary functions and use energy inequalities to study the extinction and extinction properties.Thirdly,under the premise of not adding the initial value,using the comparison principle of the variable index problem,construct the appropriate upper solution,and obtain the judgment criterion of the overall existence of the understanding.Finally,the rate estimates of many blow-up and extinction solutions and the decay estimates of global solutions are studied.This thesis consists of the following six chapters:The first and second chapters of the article first introduce the background,significance and current research status of the current research,and then give the problems,necessary preliminary knowledge and existence uniqueness theorem.In the third chapter,we give the important theorems of blow-up,extinction and global existence.In the fourth chapter,the proof of blasting correlation theorem is obtained by energy function and convexity method.In the fifth chapter,we use the comparison principle and energy inequality of ordinary differential equations to obtain the proof of extinction correlation theorem.The sixth chapter gets the result of the whole existence.It is found that variable exponents and coefficients play an important role in the Fujita blow-up of Parabolic Equations with variable exponents.