Energy Methods for Reaction - Diffusion Problems

ENERGY METHODS FOR REACTION-DIFFUSION PROBLEMS by Xing Zhong Nonlinear reaction-diffusion equations arise in many areas of applied sciences such as combustion modeling, population dynamics, chemical kinetics, etc. A fundamental problem in the studies of these equations is to understand the long time behavior of solutions of the associated Cauchy problem. These kinds of questions were originally studied in the context of combustion modeling. For suitable nonlinearity and a monotone increasing one-parameter family of initial data starting with zero data, small values of the parameter lead to extinction, whereas large values of the parameter may lead to spreading, i.e., the solution converging locally uniformly to a positive spatially independent stable steady state. A natural question is the existence of the threshold set of the parameters for which neither extinction nor spreading occurs. Even in one space dimension, this long standing question concerning threshold phenomena is far from trivial. Recent results show that if the initial data are compactly supported, then there exists a sharp transition between extinction and spreading, i.e., the threshold set contains only one point. However, these results rely in an essential way on compactly supported initial data assumption and only give limited information about the solutions when spreading occurs. In this dissertation, energy methods based on the gradient flow structure of reaction-diffusion equations are developed. The long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type is analyzed. For symmetric decreasing initial data in L(R) ∩ L∞(R), the convergence results for the considered equations are studied, and the existence of a one-to-one relation between the long time behavior of the solution and the limit value of its energy is proved. In addition, by employing a weighted energy functional, a mathematical description of the equivalence between spreading and propagation of the solutions of the considered equations is given. More precisely, if spreading occurs, then the leading and the trailing edge of the solution propagate faster than some constant speed. Therefore, if the solution spreads, it also propagates. Furthermore, for a monotone family of symmetric decreasing initial data, there exists a sharp threshold between propagation and extinction. ENERGY METHODS FOR REACTION-DIFFUSION PROBLEMS