Blow-up And Global Solutions Of Parabolic Equations With Nonlinear Boundary Flux Or Robin Boundary

Parabolic equation is an important part of partial differential equation,and there are specific mathematical models in seepage theory and combustion theory.Many natural phenomena can be described by the properties of solutions of parabolic equation,which promotes the development of partial differential equation theory.In this paper,parabolic equations with nonlinear boundary flux and Robin boundary are studied.By constructing completely different auxiliary functions,using the modified differential inequality technique and the Sobolev inequality,the existence of the global solution and the blow-up solution of the parabolic equation are studied,and the sufficient conditions are obtained to ensure the existence of the blow-up solution and the global solution of the equation.On this basis,the estimates of the upper and lower bounds of the blow-up solution are obtained.The full text is arranged as follows.In the first chapter,the research background and significance of this paper are briefly introduced,then the current state of research at home and abroad is summarized,and finally we specify the main work of this paper.In the second chapter,the blow-up solution of parabolic equation with nonlinear boundary flux is studied,the sufficient conditions for the existence of the blow up solutions are obtained by constructing proper auxiliary functions,and the upper and lower bounds of the blow-up time are given.Then we give examples to support our main conclusion.Chapter 3 discusses the upper and lower bounds of blow-up time for a class of porous media equations with Robin boundary.By constructing different auxiliary functions,the existence of global solution is guaranteed and the solution blow-ups at a finite time.In addition,we obtain the upper and lower bounds of the blow-up time when the solution blows up in high dimensional space.In chapter 4,the global solution and blowup solution of porous media equation with Robin boundary condition are discussed.By using the auxiliary function and the modified differential inequality technique,we first give the sufficient conditions of blow up of the solution at finite time and the upper bound of the blowup time,and then get the lower bound of the blowup time when the solution blows up.