Research On Well-posedness For Three Classes Of Wave Equations With Fractional Laplacian Operators

This paper studies the well-posedness of solutions for three classes of fractional Laplacian operators wave equations.Fractional Laplacian operators have important applications in finance,physics,fluid dynamics,population dynamics,image processing,minimal surfaces,and game theory.The purpose of this article is to reveal that fractional Laplacian operators with non-local characteristics on logarithmic source wave equations,generalized source degenerate Kirchhoff-type wave equations,and polynomial source degenerate and nondegenerate Kirchhoff-type dissipative wave equations effect the properties of dynamics.This work mainly uses the theory of functional analysis to study the qualitative properties of the solutions of the above equations at three different initial energy levels（subcritical,critical,and supercritical）by combining potential well theory,Galerkin method,and concave function method.And investigate and analyze the dependence on initial values.The second chapter investigates the initial boundary value problem of the fractional Laplacian operators wave equation with a logarithmic source.First of all,the chapter establishes a relationship between the norm in W₀ space and the logarithmic term according to the logarithmic Sobolev inequality,and then uses the norm to control the logarithmic term in the proof of the theorem later.Then the Galerkin method is applied to obtain the global existence of the solution at the subcritical initial energy level,and the properties of the infinite time to the solution are given by constructing a functional.Then using the scale transformation,the properties of the solution to the equation at the critical initial energy level are given.Finally,infinite time blow up of the solution of the equation at the supercritical initial energy level is given by a new initial value condition combined with a concave function method.The third chapter focuses on the initial boundary value problem of degenerate Kirchhoff-type fractional Laplacian operators wave equations with generalized source term.First of all,the related theories of related energy functional and well depth are given in the framework of potential wells.Secondly,in this chapter,a new initial value condition is found at the supercritical initial energy level.Based on the relationship between the initial energy and the initial value,the finite time blow up of the solution is obtained by the improved concave function method.In addition,when the generalized source term is a special nonlinear polynomial source term,the upper bound of the blow up time is estimated using important inequality,and the lower bound of the blow up time is given by controlling the differential inequality.The fourth chapter studies the initial boundary value problem of fractional Laplacian operators wave equations with polynomial source term,degenerate and non-degenerate Kirchhoff term,linear fractional order strong damping term,and linear and nonlinear weak damping terms.The emergence of linear fractional-order strong damping term,linear and nonlinear weak damping terms will undoubtedly make the analysis of the dynamics to the problem much more difficult.In this work,G（?）teaux derivatives are first used to deal with non-linear terms,and the existence and uniqueness of local solutions are obtained by applying the principle of contraction mapping.Then at the subcritical initial energy level,the global existence of the solution is given by the Galerkin method,the property of exponential decay of energy is given by applying differential inequality,and an appropriate functional is established to give finite-time blow up.Then the properties of the solution to the equation at the critical initial energy level are given by using the scale transformation.Finally,the finite time blow up of the solution to the equation at the supercritical initial energy level is given by the additional conditions for the initial value,and the upper and lower bounds of the blow up time are given by using the differential inequality.