Nonlinear evolution equations describe some important partial differential equations which develop over time. Large problems of these equations arise in mathematical models from physics， chemistry and biology. Thus they have a high degree of practical background. In views of mathematical theory and the development in applied sciences， it is very important to study these equations. This book will investigate the blowup phenomena and global well-posedness in nonlinear heat conduction equation and magnetohydrodynamic system， which are arose in applied sciences. The book consists of five parts in the following.
In Chapter 1， we consider the asymptotic behavior at infinity of the stationary solution which corresponding to a semilinear heat equation with exponential source， and get the behavior of the backward self-similar solution for the heat equation when approaching the blowup time， which provide convenience to study the blowup phenomenon of the singularity solution for the heat equation.
In Chapter 2， we discuss the nonexistence of type n blowup for heat equation with exponential nonlinearity in the whole space when initial data satisfy some given conditions. Using zero theory and the behavior of stationary solution of heat equation， we give a sufficient condition for the occurrence of type I blowup in the lower supercritical range by intersection comparison principle.
In Chapter 3， we study the global well-posedness for MHD system with mixed partial dissipation and magnetic diffusion in 2-dimensions. Using energy method， we consider the global well-posedness for mixed partial dissipation， we derive the desired results under some certain condition for the solutions of MHD system.
