Research On The Existence And Properties Of Solutions To The Two Types Of Parabolic Equations

This thesis investigates the existence and non-existence of solutions to a class of initial value problems for the Hénon-type parabolic equation,as well as the blow-up properties of solutions to a class of nonlinear parabolic equation systems with exponential growth.The Hénon-type parabolic equation was originally used to describe certain phenomena in celestial mechanics and is a type of partial differential equation that describes the evolution of physical systems.The nonlinear parabolic equation systems with exponential growth arises from the oscillatory behavior in chemi-cal reactions and describes the diffusion and reaction behavior of substances in chemical reactions.In the first part of this thesis,we study the local time existence and non-existence of solutions to a class of Hénon-type parabolic equations with initial values on the R~N.Firstly,a solution to a nonlinear parabolic equation with initial values is found,and the Hénon-type parabolic equation studied in this thesis is connected to it using a Cole-Hopf type transformation,obtaining an upper solution to the Hénon-type parabolic equation.Then,the local time solution of the equation is obtained using the upper solution theory and Jensen’s inequality.Secondly,under the assumption of non-existence,the initial value is estimated,and the non-existence of local time solutions is proved by contradiction.Finally,a specific application of a nonlinear parabolic equation with a nonlinear term is given.In the second part of this thesis,we study the blow-up properties of solutions to a class of nonlinear parabolic equation systems with exponential growth on a bounded domainΩ(?)R~2.Starting from initial values with energy lower than or equal to the ground state level,we use the potential well theory to prove that the solutions blow up in finite time.Specifically,the existence of solutions to the nonlinear parabolic equation system with exponential growth is first proved using the contraction mapping principle,and then the solutions of the equation system are restricted to an unstable set.The blow-up properties of the nonlinear parabolic equation system with exponential growth are proved using potential well theory.