Extinction For Some Kinds Of Reaction-diffusion Equations

Diffusion equations,as an important class of parabolic equations,come from a variety of diffusion phenomena appeared widely in nature.At the same time,it is closely linked to many fields(such as filtration,biochemistry and dynamics of biological groups).In these years,the study in this direction attracts large number of mathematicians both in China and abroad.Remarkable progress has been achieved.The study of the properties of the solutions of equations,such as the global existence,extinction in finite time,blow-up of solutions is one of the hotspot issues.This article will mainly study the extinction properties of the reaction diffu-sion equations.The extinction of the solutions of PDE has caused wide attention of the scholars since 1974.Upper and lower solution method,energy method and test function method have been used to study the extinction property of amounts of equations.There are many distinguishing practical background.In nature,in the process of biological evolution and burning of material,if the death rate is fast,or the material absorbs heat strongly,then the evolution or burning process probably does not continue,that is to say,the species extincts or combustion may stop at some time.This kind of phenomenon in mathematics is called that the solutions of these diffusion motion models extinct in finite time.There are three chapters in this thesis.First,some related concepts,in-equalities and the background of the study is introduced.Neumann and Dirich-let boundary value conditions are analyzed for the influence of the equation-s.In the second chapter,a fast diffusion equation with gradient and nonlocal source ut=?um+?|?u|q+a(?)?updx,in a bounded domain ?(?)RN(N>2)with smooth boundary(?)? is considered,where 0<m<1,q,a,p,?>0,the results of the the global existence and conditions for the extinction of solutions in finite time are established.By considering approximated problems,we prove the exis-tence,by using energy method,we prove that the solution of equation vanishes in finite time.More specially,ifm<p<1+m,m<q?2/3-m,for sufficiently small initial data,the solution vanishes in finite time.if m=q<p<1+m,q?2/3-m,for any initial data,when ?,a are sufficiently small,the weak solution vanishes in finite time.In the third chapter,a super diffusion equation with gradient and absorption terms ut=?um+?|?u|q-a(?)?updx,in a bounded domain ?(?)RN(N>2)with smooth boundary(?)? is considered,where m<-1,q,a,?>0,q<0.We also use the energy method to prove that the solution of equation vanishes in finite time.Here the absorbing items different from those in the second chapter,and the inverse Holder inequality is used.We obtain if 2/3-m?q?m/1-m,p<1+m,for sufficiently small initial data,the solution vanishes in finite time.