Global Existence,Decay And Blow-up For Two Classes Of Parabolic Equations Based On Potential Well Method

Pseudo parabolic equations and their solutions play a very important role in describing physical and other fields,such as the unidirectional propagation of nonlinear,dispersive,long waves,the aggregation of population,nonstationary processes in semiconductors in the presence of sources.In this paper,we discuss the global existence,exponential decay and blow up of solutions for semilinear pseudo-parabolic equations with cone degradation and fractional pseudo-parabolic equations with logarithmic nonlinear.The main contents are as follows:In the first Chapter,we introduce the research background and development trend of the pseudo parabolic equation,and briefly describe the main work in present thesis.In Chapter 2,we study global existence,exponential decay and finite time blow-up of solutions for a class of semilinear pseudo-parabolic equations with conical degeneration.For our purpose,we introduce a family of potential wells and its corresponding sets,and construct the relation between the existence of solution and the initial data u0 via the method of the potential wells.Then,by the usage of Faedo-Galerkin method,the concavity argument and properties of a family of potential wells,we derive a threshold result of existence and nonexistence of global weak solution:for the low initial energy case(i.e.,J(u0)<d),the solution is global in time with I(u0)>0 or ||?Bu0||L2 n/2(B)=0 and blows up in finite time with I(u0)<0;for the critical initial energy case(i.e.,J(u0)= d),the solution is global in time with I(u0)? 0 and blows up in finite time with I(u0)<0.The decay estimate of the energy functional for the global solution and the estimates of the lifespan of local solution and lower bound on blow-up time are given by making use of a differential inequality technique.In Chapter 3,we study global existence,exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity.Firstly,we recall the relationship between the fractional Laplace operator(-?)s and the fractional Sobolev space Hs and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells.Then,by the usage of Faedo-Galerkin method and properties of a family of potential wells,we derive a threshold result of existence of global weak solution:for the low initial energy J(u0)<d,the solution is global in time with I(u0)>0 or ||u0||X0(?)= 0 and blows up +? with I(u0)<0;for the critical initial energy J(u0)= d,the solution is global in time with I(u0)? 0 and blows up at +? with I(u0)<0.The decay estimate of the energy functional for the global solution is also given.The innovations of this paper are as follows:we extend similinear pseudo-parabolic e-quations from the standard Sobolev space to conical Sobolev space,and study the properties of solutions for semilinear pseudo-parabolic equations with conical degeneration;we extend pseudo-parabolic equations with logarithmic nonlinearity from the standard Sobolev space to fractional Sobolev space,and study the properties of solutions for fractional pseudo-parabolic equations with logarithmic nonlinearity.