Research Of Blow-up Behaviors For Several Nonlinear Reaction-diffusion Equations

Reaction-diffusion equations,as an important class of parabolic equations,can be used to study many problems in physics,chemistry,and biology qualitatively and quantitatively.The blow-up behaviors of the solutions are an important indicator for the stability of the models.Therefore,it is of great significance to study the blow-up behaviors for the reaction-diffusion equations.This thesis mainly studies the blow-up behaviors for three types of nonlinear reaction-diffusion equations with different source terms,including the sufficient and necessary conditions of blow-up in finite time and the estimates of blow-up time.1.The blow-up behaviors in a bounded domain are considered for a class of reaction-diffusion equation with nonlinear nonlocal heat sources and time-dependent-coefficient heat sink,under the Dirichlet,the Neumann and the Robin boundary conditions respectively.When the solution of the problem blow up in finite time,the lower bounds of blow-up time are given by constructing appropriate auxiliary functions and using Sobolev inequality,H(?)lder inequality.Finally,the feasibility of the theoretical results are verified by an example.2.The blow-up behaviors are considered for a class of porous media equation with local heat sink and nonlinear Neumann boundary condition.By constructing auxiliary functions and using Sobolev inequality,H(?)lder inequality,Young inequality,Payne and Schaefer integral inequality etc.,the sufficient and necessary conditions for blow-up in finite time are given.Moreover,if the solution blow-up in finite time,the estimates of lower and upper bounds for blow-up time are obtained.At last,the conclusions are verified by an example.3.The conditions for the global existence or the blow-up of solutions are considered for a class of nonlinear reaction-diffusion equations.By using the upper and lower solutions method and constructing self-similar functions,the global existence and blow-up in finite time of solutions are revealed.Moreover,the construction of the appropriate auxiliary functions and the application of the different inequality techniques are important keys.The estimates of the lower bounds for blow-up time are obtained in each dimensional space under Dirichlet boundary condition.