Existence And Blow-up Of Solutions For Two Types Of Nonlinear Parabolic Equations With Non-local Boundary Conditions

As one of the important research directions of nonlinear partial differential equations,nonlinear parabolic equations have a wide range of applications in the fields of natural sciences and engineering technology,and the nature of their solutions has also become a research hotspot in recent years.The research content of this paper is the global existence and blow-up properties of the solutions of several types of nonlinear parabolic equations.One type is the multilateral flow equation with non-local Neumann boundary conditions and nonlinear absorption terms,and the other type is p-Laplace equation with non-local Neumann boundary conditions and nonlinear absorption terms.For the multilateral flow equation with non-local Neumann boundary conditions and nonlinear absorption terms,the upper and lower solutions of the problem are defined first,and the comparison principle is established and proved.Then,through methods such as differential inequality techniques,construction of auxiliary functions and eigenvalue eigenfunctions,the nonlinear exponent,weight function and initial value in different ranges are demonstrated,and constructed two upper solutions,a blow-up solution in the sense of auxiliary function and a blow-up supsolution,and the two sufficient conditions for global existence and blow-up in a finite time of solutions are obtained respectively.For p-Laplace equations with non-local Neumann boundary conditions and nonlinear absorption terms,such equations are degenerate nonlinear equations,so weak solutions need to be considered.First,the weak solution of the equation is defined,then the weak upper solution and the weak lower solution are defined,and the comparison principle is established and proved.Then,under different conditions,two upper solutions and two lower blow-up solutions are constructed,and by using the principle of comparison,the two situations for global existence and blow-up in a finite time of solutions are obtained respectively.