The Study On Existence And Nonexistence Of Solutions For Several Nonlinear Partial Differential Equations

In recent years,more and more scholars have been devoted to the study of solutions for nonlinear partial differential equations,and obtained many important results on related problems.In particular,the existence and nonexistence of solutions as the important basis for studying the properties of solutions has attracted widespread attention from scholars.Accompanied by the in-depth study of nonlinear partial differential equations,fractional partial differential equations are becoming more and more important in practical application and theoretical analysis.In this paper,we study the existence and nonexistence of solutions for three classes of nonlinear partial differential equations.Firstly,in the case of three-dimensional space,we study the existence of local weak solutions for a class of nonlinear generalized Boussinesq equations.Based on the Galerkin method,by selecting an appropriate approximation solution,by using some interpolation inequalities in the Sobolev space,the energy estimation of the approximation solution is established.According to the estimation formula,the existence theorem of the local weak solution for the equation is proved.On this basis,the uniqueness theorem of the weak solution is proved by using Gronwall inequality in integral form,which generalizes the conclusions in the existing literature.Secondly,the existence and nonexistence of global solutions for nonlinear evolution equations with fractional Laplacian are studied.On the one hand,based on the test function method,by constructing an appropriate test function,the nonexistence theorem of global solutions for higher order nonlinear evolution equations in the subcritical case is proved,and the nonexistence conclusion of global solutions is extended to the correspond-ing coupled equations.On the other hand,by using Duhamel integral equivalent equation and heat kernel estimation,we prove the nonexistence theorem of global solutions for corresponding parabolic equations in the critical case and the existence theorem of global solutions in the supercritical case.The optimal Fujita critical exponent is obtained.Finally,the existence and nonexistence of global solutions for nonlinear elliptic equations with fractional Laplacian are studied.On the one hand,the nonexistence theorem of global solutions for nonlinear elliptic equations in the subcritical case is proved by using the test function method.On the other hand,by using integral equivalent equation,a priori estimate of global solutions for elliptic equations is established.According to the estimation formula,it is proved that the equation does not have positive global solutions in the critical case.In addition,the existence theorem of golbal solutions for corresponding elliptic equations in the supercritical case is proved by using the contraction mapping principle.