The Study Of Two Nonlinear Parabolic Equation (System) With Source

Nonlinear parabolic equation comes from many mathematical models in physics,chemistry,biology and economy,which has strongly practical background.On the other hand,the studying of global existence and local existence have been an important aspect in the field of nonlinear partial differential equations.This paper devotes to study the Cauchy problem for a class of degenerate parabolic system with strongly coupling source,where initial data are measures.Moreover,we examine the existence and nonexistence of solutions on the Cauchy problem for the evolutionary p-Laplace equation with nonlocal source.The thesis is composed of four chapters.In chapter 1,we summarize the background of the related problems and state the main results.In chapter 2,we study the Cauchy problem for a class of degenerate parabolic system with strongly coupling source,where initial data are measures.When the known parameters satisfy some conditions,we can overcome the difficulties which come from the interactions between the degeneracy of the principal and the strongly coupling source,and obtain the existence of solutions.Moreover,we also prove that these conditions are optimal for the existence of solutions.In chapter 3,we mainly discuss the existence and nonexistence of solutions on the Cauchy problem for the evolutionary p-Laplace equation with nonlocal source.Using the methods of priori estimates and real analysis,we establish the local existence and global existence under optimal assumptions on the initial data.What's more,a critical Fujita's type exponent is obtained.In the last chapter,we concern the summary for the whole work of this paper,as well as the prospect for the next work.