The Research On Well-Posedness For Reaction Diffusion Equation (System) With Singular Term

In this paper,we study the initial boundary value problem of a class of reaction diffusion equation with singular term and strong damping term and a class of reaction diffusion system with singular term and coupled source term.In the framework of potential well theory,this paper uses the concave function method and related functional analysis theories to study the well-posedness of three different initial levels under low energy,critical and high energy conditions and analyzes the dependence of solutions on initial values.The chapter 2 considers the well-posedness of the solution of a class of reaction diffusion equation with singular term and strong damping term under certain initial boundary conditions.Firstly,locally unique existence theorem of solutions is obtained through Galerkin method and compression mapping principle.On the basis of the existence of local solutions,this chapter studies the global existence and asymptotic behavior of solutions under the condition ofJ?u₀??d In addition,it is obtained that solutions blow up in finite time and the upper bound of the blowup time is estimated.The global existence,asymptotic behavior and finite time blowup of solutions are proved WhenJ?u₀?=d.Finally,the topic obtains the finite time blowup of solutions by constructing a new energy functional and combining with Hardy inequality in the case of.J?u₀??dFurthermore,based on a single reaction diffusion equation with singular term,the initial boundary value problem of a class of reaction diffusion system with singular term and coupled source term is studied in Chapter 3.Firstly,due to the symmetry of coupling source term,this chapter gives potential energy functional,Nehari flow,potential well depth and some basic lemmas.Then,this chapter proves the invariant sets of solutions according to lemmas.Under low initial energyJ?u₀,v₀??dand critical initial energyJ?u₀,v₀?=d?respectively,the global existence,asymptotic behavior and finite time blowup of solutions are obtained in this chapter.Finally,this chapter is also known for the finite time blowup of solutions at the high initial energy levelJ?u₀,v₀??d?by the comparison principle.