Research On Well-posedness For Two Class Of Nonlinear Evolution Equations With Variable Exponents Laplacian

This thesis pays attention to the global well m-posedness of the solution to m-Laplacian and m(x)-Laplacian parabolic equations by using the potential well method,concave method and functional analysis theory.Further analyzes the solution with respect to the dependance of the initial value.In this thesis,we study the global well-posedness of solution for above two equations at subcritical initial energy,critical initial energy and supercritical initial energy respectively.The application of potential well theory in the parabolic equation with variable exponential Laplacian is extended by studying two classes of nonlinear parabolic equations with variable exponential Laplacian.The potential well theory is further developed and improved.The second chapter studies the initial boundary value problem for a nonlinear parabolic equation with $m$ m(x)-Laplacian.The origin model of the-Laplacian equation can be regarded as a comparison between the dynamical behavior of the model with nonlinear diffusion and the evolution property associated with the linear diffusion model.We prove the blowup in finite time of the solution and estimate the upper and lower bounds of the blowup time at subcritical initial energy with the help of two different auxiliary functions.Further,we parallelly extend all obtained results in subcritical initial energy to critical initial energy by scaling transformation of initial data.In super-critical initial energy,since the energy is no longer controlled by the depth of potential well,which implies that the invariant set is invalid.By imposing a restriction related to initial value on initial energy,we get the invariant set.Combining the concave function method,we prove the finite time blowup of the solution and estimate the lower and upper bounds of the blowup time by introducing two different auxiliary functions.The research meaning of this chapter is that the well-posedness of solution of the initial boundary value problem for nonlinear parabolic equation with $m$-Laplacian is comprehensively studied at the full energy level.The third chapter considers the well-posedness of solution for a nonlinear parabolic equation with m(x)-Laplacian at three different initial energy level.This equation is used to describe the flow of electrorheological fluids.We establish the global existence and uniqueness of solution with the help of the Galerkin methods and the boundedness theorem at sub-critical and critical initial energy level.We also prove the asymptotic behavior of solution by analyzing the relationship between the potential energy functional and the Nehari functional,and the finite time blowup of solution by analyzing the relationship between the depth of potential well and the norm of solution by using the method of concave function.Based on this,we respectively estimate the blowup time by introducing two different auxiliary function.Furthermore,we prove the blowup in finite time of solution and estimate the lower and upper bound of the blowup time at super-critical initial energy level.The research meaning of this chapter is to systematically and structurally study the global well-posedness of solution of the initial boundary value problem for nonlinear parabolic equation with $m(x)$-Laplacian.