Research On Well-posedness For A Class Of Noinear Evolution Equations With Singular Potential

By using the method of potential well,functional analysis and energy estimation,this paper focuses on the well-posedness for a class of nonlinear parabolic equation with singular potential and logarithmic nonlinearity in sobolev space,combing the method of Galerkin approximations with concavity function,in order to explore the effect of initial data,sigular potential and logarithmic non-linearity to the properties of solution in nonlinear evolution equation.Under the different circumstances of sub-critical energy and critical energy,we obtained the result of the existence of the global Solution and blow-up phenomena,improving further the application of the theory of potential well.In Chapter 1,we introduce the research background of nonlinear parabolic equation with singular potential and the existing achivement to draw out our research purpose.In Chapter 2,we present some knowledge involved in real analysis and functional analysis.Forp-Laplacian operator,we consider two case:p=2 and p>2.When p=2,Δp=Δis a linear operatot and when p>2,Δp is a nonlinearity operator.In Chapter 3,we pay attention to a class of linear parabolic equation with singular potential and logarithmic nonlinearity,whenp=2.First of all,on the basis of potential well and the Galerkin method,we prove the existence of the global solution in situations of sub-critical energy and critical energy.What’s more,under the circumstance of the existence of the global solution,we prove the phenomenon of infinite time blow up of solution.In Chapter 4,we focus on a class of nonlinear parabolic equation with singular potential and logarithmic nonlinearity,whenp>2.Firstly,similar to approaches in the chapter three,by using the Sobolev inequality and Galerkin method,we obtain the local existence of weak solution.Secondly,we show the existence of the global solution in situations of sub-critical energy and critical energy by potential weill.Then,we derive a sufficient condition that the weak solution converges to the stationary solution when time tends to infinity.Finally,finite time blow up of solution will be proved in sub-critical energy by constructing auxiliary functions and the concavity method.