Research On Well-posedness For Coupled Parabolic Systems On A Manifold With Conical Singularities

This paper mainly uses the potential well method to study the well-posedness of the initialboundary value problem of coupled parabolic systems with exponential source terms on a manifold with conical singularities.This problem is related to the ignition model for the thermal explosion of two mixed solid fuels with a limit degree.The initial state of solid fuel has a certain influence on the phenomena during the explosion moment after ignition and after the explosion.Therefore,this paper focuses on the influence of initial value on the dynamical behavior of the solution of this considered problem.This paper considers both the coupling relationship of nonlinear sources and the non-smoothness in regional space.In the case of the two actings simultaneously,the global existence and finite time blow-up of the corresponding solutions are obtained for three different initial energy levels,in order to reveal the combined effect of the coupling source,space,and initial value on the well-posedness of the solution.The second chapter of this paper mainly introduces the nature of the cone Sobolev space where the problem lies and related inequalities on it,such as H(?)lder's inequality,Sobolev inequality,Poincaré inequality,and exponential inequalities on a manifold with conical singularities.At the same time,this chapter also introduces the potential energy functional and the Nehari functional and studies their properties.In the third chapter of this paper,by using the potential well method,Galerkin method,exponential inequality on a manifold with conical singularities,and concave function method,the global existence and finite-time blow-up of solutions for the initial-boundary value problem of coupled parabolic equations with exponential source terms on a manifold with conical singularities under subcritical initial energy levels and critical initial energy levels are derived.The fourth chapter of this paper obtains the finite time blow-up and the upper bound of the blow-up time of the solution for the initial-boundary value problem of the coupled parabolic equations with exponential sources on a manifold with conical singularities under arbitrarily positive initial energy level by concave function method and differential control inequality.