Asymptotic Behavior Of Solutions To Several Classes Of Semilinear Parabolic Equations And Systems

This thesis investigates the global existence and blowing-up properties of solutions to several classes of semilinear parabolic equations and systems.The thesis is divided into two parts.The first part considers the asymptotic behavior of the solutions to semilinear parabolic equations and systems with degenerate diffusion coefficients.The second part deals with the Cauchy problem of semilinear parabolic equations and systems with general gradient terms.In the first part,we consider the semilinear parabolic equations and systems with degenerate diffusion coefficients.We divide the first part into three chapters to introduce the influence of the degenerate diffusion coefficients on the large-time behavior of solutions of equations and systems.The first chapter studies the global existence and blowing-up properties of solutions to one-dimensional semilinear parabolic equations with general diffusion coefficients.We use the method of constructing self-similar solutions to prove the global existence of solutions and we apply the methods of weighted energy estimates to prove the blowing-up property of nontrivial solutions.The generality of the degenerate diffusion coefficient makes it necessary to select an appropriate weight function in the energy estimates method,and the constructed self-similar solution has more complex structures.In particular,for the problem in a bounded domain,the nontrivial solution must blow up in a finite time if the degeneracy is strong enough.For the problem in an unbounded domain,we choose appropriate weights and apply the methods of weighted energy estimates to determine the interaction of the degenerate diffusions and the reactions,so that we could calculate the critical Fujita exponent and establish blowing-up theorems of Fujita type.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 second chapter investigates the asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with degenerate diffusion coefficients.For the problem in a bounded domain,it is proved that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large one 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 apply the methods of weighted energy estimates to determine the interaction of the degenerate diffusions and the reactions,and obtain the critical Fujita curve.Furthermore,blowing-up theorems of Fujita type are established.It is also shown that the critical Fujita curve is determined by the strength of degeneracy.In particular,it is finite if the degeneracy is not strong,while is infinite.In the end of the chapter,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 third chapter,we discuss the asymptotic behavior of solutions to coupled semilinear degenerate equations with general diffusion coefficients.This chapter,based on the previous two chapters,is to study the influence of the degeneracy degree of the more general diffusion coefficient and the coupled system on the large time behavior of the solutions.Since the system we discussed do not have self-similar structure,it is necessary to construct the supersolution of the coupled system with self-similar structure.At the same time,the generality of diffusion coefficient makes it more difficult to deal with the global solution of the problem.It is still necessary to select proper energy weight function to overcome the difficulty for energy estimates caused by the existence of general degenerate diffusion coefficients,and the selection of energy weight function is more complicated because of the generality of the diffusion coefficient.Since the problems in this paper are on coupled parabolic systems with boundary degeneracy,some complicated estimation are needed.In the end of the chapter,it is also proved by analyzing the properties of solutions that any nontrivial solution must blow up in a finite time in the critical case.In the second part,we consider the Cauchy problem of semilinear parabolic equations and systems with a general gradient term.We divide the second part into two chapters to introduce the influence of the gradient term coefficients on the large-time behavior of solutions of equations.The fourth chapter studies the Cauchy problem of semilinear parabolic equations with general gradient terms,and discusses the influence of the diffusion and gradient term coefficients on the global existence and blowing-up properties of solutions.To prove the blowing-up of solutions,we apply the methods of weighted energy estimates and establish blowing-up theorems of Fujita type.And we obtain the critical exponent is determined by the diffusion coefficients and the behavior of the coefficients of gradient terms at the positive and negative infinity.For the global existence of nontrivial solutions,we construct global self-similar supersolutions although does not possess a self-similar structure.Since the gradient term coefficient does not have symmetry,it is necessary to select an appropriate weight function in the energy estimation method,and a more complex structure is required when constructing the self-similar solution.Furthermore,since the effect of the gradient term on the large behavior of the solutions is not at the same level as the diffusion term,we have to do many complicated and precise calculations in this thesis.For the critical case,since it is not possible to intuitively and clearly compare the energy relationship between the 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.The fifth chapter studies the Cauchy problem of semilinear parabolic systems with a general gradient term.We apply the methods of weighted energy estimates and construct selfsimilar supersolutions to investigate the influence of the diffusion and gradient term coefficients on the global existence and blowing-up properties of solutions.We obtain that the critical Fujita curve is determined by diffusion coefficients,gradient term coefficients and nonlinear source indexes.The first difficulty in this chapter is that the effect of the gradient term on the large behavior of the solution is not at the same level as the diffusion term,and the coupled systems need to be scaled during energy integration.Second,the gradient term coefficient is not symmetrical,and it is necessary to select an appropriate weight function and construct appropriate self-similar supersolutions.This thesis studies the global existence and blowing-up properties of solutions to semilinear parabolic equations and systems with degenerate diffusion coefficients and general convection coefficients,respectively,which displays how the degenerate diffusions,the convections and the reactions influence the large time behavior of solutions.We prove the global existence of nontrivial solutions by constructing suitable self-similar supersolutions.We prove the blowing-up properties of nontrivial solutions through the methods of weighted energy estimates,and we obtain the critical exponent/curve of the related problem.Moreover,blowing-up theorems of Fujita type are established.