The Research On Well-posedness For Three Classes Of Nonlinear Diffusion Equations

This paper is aim to reveal the affect of the initial data, the coupled nonlinear source term and the special medium void as well as the weight cone operator on the global existence and finite time blowup of solutions for three classes of nonlinear diffusion equation with the coupled nonlinear source terms, the doubly degenerate system with medium void on weight cone operator by using the potential well method, comparison principle and functional analysis together with the potential well method.Chapter 2 studies a class of Lotka-Volterra cooperating reaction-diffusion system. By constructing a linear system relate to this problem and using the classical comparison principle we proved the local existence of the Lotka-Volterra system. And then by reducing the order of this system and constructing the approximate solutions we obtain global existence of solutions.Moreover,by introducing a auxiliary function and starting from the classical analysis method,we show the blowup result of solutions. In this part, we study the directed diffusion with coupled source term and reveal the relationship between the nonlinear diffusion term and the nonlinear source term.Chapter 3 focuses on the global existence and finite time blowup of solutions for a class doubly degenerate equation with special medium void at low initial energy level. By applying the concavity method and the potential well this chapter shows the sharp condition of the existence and non-existence of global solutions. In detail, we obtain the existence of global solutions at low initial energy level by constructing an approximate solutions. This chapter makes the first try to consider the global existence and finite time blow-up of solutions for this system,which generalize the obtained result on the system.Chapter 4 considers the global existence and finite time blowup of solutions for a class of nonlinear parabolic equation on the weight conical degenerate manifold at the low initial en-ergy level and critical initial energy level. Since the weight conical manifold is a new topology manifold, the research about their properties of the different functionals is always considered by many authors. However, there are so many difficulties on the energy functional for this system and the potential well for the weight conical manifold, because of the complicated nature of functionals and special functional definitions of the weight conical manifold. Ac-cording to the structure of conical manifold, we introduce the weight conical Hilbert space and the potential well method with the weight conical manifold. At last, we obtain the sharp condition of global existence and blow up of solutions the low energy level, and then by using the scaling technology, we get the well-posedness of solutions at critical energy. This sec-tion considers global well-posedness of solutions for the nonlinear parabolic with the weight conical manifold.