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

This thesis studies the existence of solutions of the two types of parabolic equations.On one hand,the existence of solutions to a class of parabolic inequality equations is considered;on the other hand,the existence of solutions to Hénon type parabolic equation with initial values on bounded regions is studied.The first part of this thesis studies the existence of non-trivial non-negative solutions and positive solutions for a class of parabolic inequality equations in the whole space.Firstly,the non-existence of positive solutions is obtained by the Young’s inequality and the optimal Liouville results of a single parabolic inequality.Secondly,the non-existence of non-trivial non-negative solutions is obtained by using the strong maximum principle and the upper and lower bounds of the solution.Finally,we prove the existence of the positive solution by using the properties of the parabolic norm,the inverse H(?)lder’s inequality and the properties of the heat core.The second part of this thesis investigates the existence of classical solutions to two-dimensional Hénon-type parabolic equations which nonlinear term has an exponential growth.By comparing the initial value and the singular solution,we can obtain three different cases of the solution.Firstly,when the initial value is smaller than the singular solution,the existence of the classical solution of the corresponding elliptic equation is proved by the shooting method,and then a singular local time solution of the equation is obtained by the compression mapping principle.Secondly,when the initial value is equal to the singular solution,the singular solution is a solution of the equation,and a classical solution of the equation is obtained by applying the Cole-Hopf transformation and the Perron’s monotone method.Finally,when the initial value is larger than the singular solution,we prove the non-existence of the non-negative solution by using contradiction and the estimation of the upper and lower bounds of the initial value.