On Potential Wells And Applications To Some Nonlinear Parabolic Equations

Combining potential well method and some differential inequality techniques,in this dissertation,we consider the asymptotic properties for some nonlinear parabolic equations,such as globally existence,blow-up in finite time,vacuum isolating phe-nomena and so on.Firstly,we discuss the solution to a semilinear pseudo-parabolic.Under the low initial energy case,we find a explicit vacuum region.For the arbi-trary high initial energy case,we get a initial condition such that the corresponding solution blow up in finite time,and then we estimate the blow-up time form upper.Moreover,we prove that there exists blow-up solution with arbitrary high initial en-ergy.Secondly,we consider a four order nonlinear parabolic with the Hessian term.Our results shows that the solution will blow up in finite time with L~2-norm when initial energy is negative,and then we obtain the upper bound estimates of blow-up time and blow-up rate.Furthermore,we also study the solution with high initial energy.The initial conditions for solution blows up in finite time and exists glob-ally are established,respectively.Finally,we discuss the properties of solution to a non-Newton polytropic filtration equation under critical and high initial energy.It should be pointed out that we also get a extinction and non-extinction result when the solution own critical initial energy.Concretely speaking,there are following four chapters in this dissertation:In the first chapter,we first introduce briefly the origin and development of potential well method during recent years.Then we show the study background and existing results of the 3 parabolic equations,which will be considered later in this paper.Based on those introduction,we give the innovations and the methods of this dissertation.In the second chapter,we study initial and homogeneous Dirichlet boundary value problem of a semilinear pseudo-parabolic equation.Firstly,when the initial energy is positive and bounded form upper by potential well depth,we get a explicit vacuum region,which is an annulus,that is to say,all solution will isolate by the annulus.In inner of the annulus(we call it "global region"),the solution exists globally,and the solution will blow up in finite time if they belong to the annulus'exterior(we call it "blow-up region").When the initial energy is non-positive,we find another vacuum region,which is an ball,and in this case,the "global region"disappear,all solution will blow up in finite time.Secondly,we obtain a sufficient condition such that the solution with arbitrary high initial energy blow up in finite time,and then we get a upper bound estimate for blow-up time.Finally,we prove that there exists some initial value,which ensure that the corresponding solution with arbitrary high initial energy blow up in finite time.In the third chapter,under the homogeneous Dirichlet boundary condition,we continue to consider a forth order parabolic equation with the Hessian term.Based on the existing results,we study the two open questions,that is Lp-norm blow-up and the dynamics behavior of solution with high initial energy.In the forth chapter,we consider initial and homogeneous Dirichlet boundary value problem of a non-Newton polytropic filtration equation.We get the initial conditions such that the solution exist globally and blow up in finite time under critical and high initial energy cases,respectively.For the critical initial energy case,we also show that the global solution will decays exponentially,and we get a extinction and non-extinction result.As far as the high initial energy case is concerned,we prove that the solution exists globally under some proper assumptions,and the solution decreasing to 0 as time tends to infinite.Moreover,we get a initial condition such that the solution with high initial energy will blow up in finite time or at infinite.