| This article mainly establish the Fujita type theorems of several quasilinear cou-pled diffusion equations,describe the large time behavior of the corresponding solution and then obtain the critical Fujita exponent of the problem.Compared with other types of coupled systems,the Fujita type theorem of slow diffusion coupling system of achievement is relatively few.This paper mainly discusses four kinds of slow diffusion coupling systems.In the first chapter,we adopt the inte-gral estimation method to prove that any nontrivial solution of the quasilinear diffusion equations Cauchy problem in a finite time blow-up.The advantage of integral estima-tion is that it intuitively and clearly compares the energy relationship between diffusion terms,reaction terms and other factors in the equations,and obtain the influence of various factors on the large time behavior of the solution.For the blow-up property of the problem solution,we use the method of constructing the selfsimilar solution and the comparison principle to prove our conclusion.At the end of this chapter we analyze the global existence and blow-up properties of solutions in critical case.By the above discussion,we get the critical Fujita exponent of the problem and depict the diffusion reaction term to the large time behavior of solutions,so as to establish the complete the problem Fujita type theorem.In the second chapter,we consider more general Cauchy problem for coupled quasilinear reaction diffusion systems.We use the energy estimation and comparison principles to study the global existence and blow-up properties of solutions to the problem of equations coupled by reaction terms.In addition,we also talk about the large time behavior of the critical case being the solution to the problem.Ultimately we obtain the critical Fujita exponent of the problem,describe the influence of each element of the problem on the large time behavior of the solution and then we establish the complete Fujita type theorem.The difficulty in this chapter is that the reaction term is related to the spatial position and the equations of self-similar supersolutions have more complex structures.Hence,we need a more precise and meticulous integral estimation to prove our conclusion when making integral estimates.In the third chapter,we study a coupled quasilinear convection diffusion equations.Since the convection term of such a system is odd at the origin,we consider the problem of the homogeneous Neumann outer region problem of the system.The main purpose of this chapter is to study the effect of the convection term on the large time behavior of the solution.When we study the blow-up property of the solution,the convection term makes the process of energy estimation more complicated.The form of convection term in the equations is exactly the same,so we can still choose the same type of weight function with the same support set to make our estimation.Besides,the difference of reaction term makes it necessary to make the two energy integrals reach the same order of growth through telescopic transformation.When proving the global existence of the solution of the problem,we notice that convection term does not complicate the structure of the self-similar supersolutions of the equations from similarity complicated.However,it will make our construction process more complicated.Our work shows that the influence of convection term on the large time behavior of the solution is at the same level as that of diffusion term and reaction term,which will directly affect the critical Fujita exponent of the problem and make the critical Fujita exponent infinite under certain conditions.In the fourth chapter,we consider a more general homogeneous Neumann outer region problem for coupled quasilinear convection diffusion equations.On the basis of chapter 3,we consider quasilinear coupled reaction convection diffusion equations with different convection term coefficients.We focus on how the critical Fujita exponent of the problem with different convection term coefficients will change.The difficulty in this chapter lies in the fact that the same type of weight function cannot meet our needs due to the different coefficients of the convection term.We need to select different types of weight functions with the same support set to make our estimation according to the nature of the convection term.In addition,different convection terms make our integral estimation really complicated and need to be classified and discussed according to different conditions.At the end of this chapter,we also prove that the critical case of the problem belongs to the blow-up case,that is,any nontrivial solution of the problem blows up at a finite time in the critical caseIn this paper,we study the global existence and blow-up properties of the solutions of four coupled quasilinear diffusion equations.We reveal the influence of diffusion term,reaction term,convection term and other factors on the large time behavior of the solution of the problem.We obtain the critical Fujita exponent of the solution of the problem and establish the Fujita type theorem of the problem. |
