Asymptotic Behavior Of Solutions To Quasilinear Diffusion Equations And Systems Degenerating On The Boundary

This thesis aims to investigate the global existence and blowing-up properties of solutions to quasilinear diffusion equations and systems degenerating on the boundary by the methods of weighted energy estimates and constructing self-similar solutions.The quantitative relationship between the critical Fujita exponent/curve and the degenerate diffusion coefficient,the nonlinear diffusion exponent and the nonlinear source term is explored,and the Fujita theorem is established.In the first part,we consider a class of quasilinear slow diffusion equations and fast diffusion equations with degenerate diffusion coefficient of power function type.We divide the first part into two chapters to introduce the influence of the diffusion speed on the large time behavior of solutions of equations.The first chapter studies the global existence and blowing-up properties of solutions to a class of quasilinear slow diffusion equations with degenerate diffusion coefficient of power function type.In this chapter,the method of energy estimates is used instead of the method of constructing blowing-up solutions to prove the blowing-up properties of solutions.The advantage of integral estimation is that it intuitively and clearly compares the energy relationship between degenerate diffusion terms,reaction terms and other factors in the equations by selecting appropriate weight function,and obtain the influence of various factors on the large time behavior of the solution.We use the method of constructing self-similar solutions and the comparative principle to prove the global existence of solutions.In the second chapter,we discuss the asymptotic behavior of solutions to a class of quasilinear fast diffusion equations with degenerate diffusion coefficient of power function type.We proved to have same conclusions as in the slow diffusion case.For the problem in a bounded domain,it is proved that there exist both nontrivial global solutions and blowing-up solutions if the degeneracy is not strong,while any nontrivial solution must blow up in a finite time if the degeneracy is strong enough.For the problem in an unbounded domain,we obtain the critical Fujita exponent and establish blowing-up theorems of Fujita type.It is proved that the critical Fujita exponent determined by the degree of the degeneracy of the diffusion coefficient on the boundary and the exponents of the nonlinear diffusion.For the critical case,we analyze the global existence and blowing-up properties of solutions by a series of elaborate energy estimates.And it is shown that it belongs to the blowing-up case for any nontrivial solutions.In the case of fast diffusion,we need to construct different weight functions and self-similar solutions according to the degree of diffusion compared with that in the case of slow diffusion.The second part investigates quasilinear slow diffusion equations and fast diffusion equations with general degenerate diffusion coefficients.In two chapters,the general degenerate diffusion coefficient and the influence of diffusion speed on the large time behavior of solutions are introduced.In the third chapter,we consider the asymptotic behavior of solutions to quasilinear slow diffusion equations with general degenerate diffusion coefficients.This chapter aims to discuss the effects of more general degenerate diffusion coefficients,nonlinear diffusion exponents and nonlinear sources on the global existence and blowing-up properties of solutions.This chapter still adopts the method of energy estimates to prove the blowing-up properties of solutions.Due to the generality of the degenerate diffusion coefficients of equations,it is necessary to select the more complicated weight function.Furthermore,the existence of nonlinear diffusion exponent requires more complex estimates.For the critical case,since it seems to be not intuitive and clear to compare the degree of influence among the degenerate diffusions,reactions and other factors in the equations,we need to analyze the properties of the solutions in the critical case and prove the blowing-up properties of the solutions through a series of more accurate and detailed energy estimates than before.The fourth chapter studies quasilinear fast diffusion equations.We discuss the large time behavior of solutions to quasilinear equations with more general degenerate diffusion coefficients in the case of fast diffusion.Similar to the method of slow diffusion case,this chapter proves the global existence and blowing-up properties of solutions through the method of energy estimates and the method of constructing self-similar solutions.Due to the speed of diffusion,it is necessary to select the appropriate weight function and self-similar solutions to obtain the asymptotic behavior of solutions.And the critical Fujita exponent is determined by the degree of the degeneracy of the diffusion coefficient on the boundary,the nonlinear diffusion exponent and the nonlinear source.In the third part,we consider a class of quasilinear fast diffusion system with degenerate diffusion coefficient of power function type.This chapter is to study the influence of the coupled system on the large time behavior of the solutions.Since the system we are discussing does not have self-similar structure and has a nonlinear diffusion exponent,it is necessary to construct the supersolution of the coupled system with self-similar structure.Using the energy estimates to prove the blowing-up properties of solutions,since the nonlinear sources have different exponents,we need to make the two energy integrals reach the same order of growth through telescopic transformation.Since this part considers the related problems of coupled equations,more complex estimates are required throughout the chapter.In this thesis,we study the global existence and blowing-up properties of solutions to quasilinear diffusion equations and systems degenerating on the boundary.It is revealed that the influence of the degenerate diffusion coefficient on the boundary,the nonlinear diffusion exponent and the nonlinear source on the large time behavior of solutions to the problem.The critical Fujita exponent/curve of the problem is obtained.Moreover,blowing-up theorems of Fujita type are established.