| This thesis study the global existence and blowing-up properties of solutions to the Cauchy problem of a class of multi-dimensional semilinear parabolic equations with general diffusion coefficients.The thesis is divided into four chapters.The first chapter is an introduction,which introduces the study process and obtained results of the asymptotic behavior of non-degenerate parabolic equations to degenerate semilinear parabolic equation solutions,and points out the differences between this study and previous studies.Chapter 2 is a preliminary knowledge,introducing the definition of solutions to the Cauchy problem of a class of multi-dimensional semilinear parabolic equations with general diffusion coefficients,well-posedness of weak solution and comparative principle.The third chapter establishes blowing-up theorems of Fujita type.We apply the methods of weighted energy estimates to prove the blowing-up property of nontrivial solutions.The generality of the diffusion coefficient and spacial dimensions make it necessary to select an appropriate weight function in the energy estimates method.We use weighted energy estimates to determine the interaction of the diffusions and the reactions,so that we could calculate the critical Fujita exponent and establish blowing-up theorems of Fujita type.It is shown that the critical Fujita exponents are determined by the spacial dimensions and the asymptotic behavior of the diffusion coefficients at infinity.We use the method of constructing self-similar solutions to prove the global existence of solutions,and due to the generality of the diffusion coefficient and spacial dimensions,the constructed self-similar solution has more complex structures.Chapter 4 considers the critical case,since the critical case diffusion term and the reaction term are at the same order,we analyze the properties of the critical case solution to find that the critical case belongs to the blowing-up case.Furthermore,the critical case is proved to belong to the blowing-up case. |
