Radon Measure-Valued Solutions For A Class Of Nonlinear Degenerate Parabolic Equations

In the last fifty years,nonlinear partial differential equations with measure data,especially nonlinear elliptic and parabolic equations with measure data have become the main subject of theoretical research on partial differential equations with measure data.The interest in this area is theoretical,due to its deep relations to other branches of mathematics,such as probability theory,harmonic analysis,and differential equations.On the other hand,the kind of problem study in this dissertation has an important theoretical significance and applications,because their equations come from many fields,for instance astrophysics,thermistor problems,branching processes and superdiffusions.This work mainly studies the existence,uniqueness,qualitative properties,decay estimates and asymptotic behavior of Radon measure solutions to the degenerate parabolic equations.The main results are given as follows:In the first part,we prove the existence,uniqueness,decay estimates and asymptotic behavior of Radon measure solutions for a class of nonlinear parabolic equations involving the linear inhomogeneous heat equation solution as a source term under Neumann boundary condition and bounded Radon measure as initial data.To attain this,we prove that the solution of the inhomogeneous linear heat equation under Neumann boundary condition with measure data has a unique Radon measure-valued solutions.Then,we prove the existence and unique solutions of the degenerate parabolic problem by using the approximation method and compactness theorem in BV space.Furthermore,we construct the suitable function and take it as a test function in the approximation problem,and combining some measure properties,then the decay estimate of the Radon measure-valued solutions is obtained.In addition,we construct a pseudo-steady-state problem and we establish the decay estimate of the Radon measure solution of the difference between the degenerate parabolic problem and the pseudo-steady-state problem.Then the asymptotic behavior of Radon measure-valued solutions for a large-time is deduced.In the second part,the Dirichlet boundary problem for nonlinear degenerate parabolic equations is studied.To prove the existence and uniqueness of Radon measure-valued solutions,we employ a similar method to the previous part.In order to analyze the asymptotic behavior of Radon measure solutions,we construct the pseudo-steady-state problem of the Dirichlet boundary problem for nonlinear degenerate parabolic equations such that the pseudo-steadystate problem has a Radon measure solution.The decay estimate of the Radon measure-valued solution of the difference between nonlinear degenerate parabolic equations with homogeneous Dirichlet boundary and the corresponding pseudo-steady-state problem is established.Based on this estimate,the asymptotic behavior of Radon measure solution of the Dirichlet boundary problem for nonlinear degenerate parabolic equations is obtained.In the third part,we discuss the nonlinear parabolic equations with Neumann boundary condition with Radon measure as the initial data.The existence of Radon measure-valued solution is proved by using the approximation method and compactness theorem in BV space.Then,by constructing the suitable auxiliary functions and using them as test functions in nonlinear parabolic regularization problems with both source terms and Neumann boundary conditions,we establish several important inequalities and prove the decay estimates of Radon measurevalued solutions with additional inequalities.