Research On The Existence Of Self-similar Solutions For Several Types Of Nonlinear Parabolic Equations

Nonlinear parabolic equation is one of the important equations of nonlinear partial differential equations.It has been widely used in physics,chemistry,biology,control and other fields,and has become one of the hot topics for many scholars in recent years.In this thesis,we mainly study the existence of positive self-similar solutions or sign-changing self-similar solutions of parabolic equation and parabolic systems under appropriate initial conditions by applying the shooting method,the principle of compressed mapping,perturbation theory,etc.On the one hand,this thesis studies the existence of positive self-similar solutions and sign-changing self-similar solutions of Hénon parabolic equation.On the other hand,we study the existence of positive self-similar solutions for two types of parabolic systems.Compared with the previous research results,more abundant results on the existence and properties of solutions for Hénon parabolic equation and two types of parabolic systems are obtained in this thesis.In the first part of this thesis,we prove the the existence of positive self-similar solutions and sign-changing self-similar solutions of Hénon parabolic equation.First,Hénon parabolic equation can be transformed into ordinary differential equation through transformation.The global existence and uniqueness of solutions for ordinary differential equation are proved by using the principle of contractive mapping and the energy of the solution,and the asymptotic behavior of the solution is obtained by iteration.Second,by using the existence and properties of positive solutions of ordinary differential equation,we can obtain that Hénon parabolic equation have positive solutions when the initial value is zero.In order to study the existence of positive solutions and sign-changing solutions of Hénon parabolic equation under the condition of singular initial values,it is also necessary to transform ordinary differential equation.Third,we prove the existence and uniqueness of the solution of the converted ordinary differential equation under appropriate conditions by using the principle of compressed mapping.Finally,the shooting method can be used to prove that,under the singular initial value conditions,the Hénon parabolic equation has positive self-similar solutions and sign-changing self-similar solutions.In the second part of this thesis,we study the existence of positive self-similar solutions for two types of parabolic systems.First,we transform it into a system of ordinary differential equations by transformation,and then we prove the existence and properties of solutions of the system of ordinary differential equations by using the principle of contractive mapping.Second,it can be obtained that,under appropriate initial conditions,the parabolic system have positive self-similar solutions by using the properties and transformations of solutions of ordinary differential equations.Finally,by using perturbation theory,we can obtain the existence of positive self-similar solutions of parabolic system with a wider range of initial conditions.