Study On The Singularities For The Solutions To Some Nonlinear Evolution Equations

In this dissertation,we consider the singularities of solutions to several nonlinear evolu-tion equations.Firstly,we consider a class of parabolic equations with singular potential term and a class of semi-linear edge degenerate parabolic equations with singular potential term.By constructing a new functional which falls between the energy functional and the Nehari function-al,we obtain a new blowup condition,which demonstrates the possibility of finite time blowup when the initial energy is larger than the critical initial energy.Then we study two classes of p-biharmonic equations.Under appropriate conditions,we prove the global existence of weak solutions by means of the Galerkin approximate method.And then through some inequalities of nonnegative functions,we get the results on blowup,extinction and non-extinction of weak solutions to the models.There are five chapters in this dissertation:In the first chapter,we mainly introduce the current researches about the blowup and the extinction of solutions to nonlinear evolution equations,as well as the purposes,the innovations and the methods of this dissertation.In the second chapter,we study the blowup of a class of nonlinear parabolic equations with singular potential term.In this chapter,we not only prove that the solution of the equations blows up when the initial energy less than the critical initial energy,but also give the result on the blowup of solutions with the condition that the initial energy not less than the critical initial energy.In the third chapter,we continue to study the blowup of weak solutions to a class of semi-linear edge degenerate parabolic equations with singular potential term.We firstly introduce the basic theory of the edge type weighed p-Sobolev space and some lemmas,then we prove that the weak solution of the equations blows up in finite time when the initial energy larger than the critical initial energy.In the forth chapter,we consider the properties of weak solutions to a class of nonlocal p-biharmonic equations with the nonlocal term |u|q-|?|-1 ??|u|qdx when p?2.We not only prove the blowup of weak solutions with the initial energy E(u0)?0,but also the properties of extinction and non-extinction with appropriate initial values.In the fifth chapter,we study the properties of weak solutions to a class of nonlocal p-biharmonic equations with the nonlocal term |u|q-1 u-|?|-1 ??|u|q-1 udx when p>2.We discuss the global existence and the blowup of solutions in two cases of E(u0)?0 and E(u0)>0,as well as the properties of extinction and non-extinction of weak solutions.Finally,we estimate the upper bound to the blowup time.