| This paper mainly uses the potential well theory,Galerkin method,concavity method and variational principle to study the existence and blow up of weak solutions of three kinds of nonlinear reaction-diffusion systems.For a class of p(x)Laplacian equations with logarithmic sources,the local existence and finite time blow up of weak solutions are obtained by Galerkin method and concavity method,respectively.In addition,by means of potential well theory and Nehari manifolds,the global existence of the weak solution of the equation under subcritical state(J(u0)<d)of initial energy is given.For a class of p-Laplacian equations with logarithmic sources and singular terms,the finite-time blow up of weak solutions under subcritical and critical states(J(u0)≤d)is obtained by using the improved concavity method under appropriate conditions.In addition,by means of variational principle,the sufficient conditions for the global existence and blow up of the weak solutions of the equation in the supercritical state(J(u0)>d)are given.Finally,the asymptotic state of the weak solution is given,that is,when t→+∞,the weak solution converges to the steady state solution of the equation.For a class of Laplacian systems with logarithmic sources.By using Galerkin’s method,potential well theory and concavity method,the global existence of weak solutions and finite time blow up under subcritical states(J(u0,v0)<d)and critical states(J(u0,v0)=d)are discussed respectively.Then,by means of variational principle,the sufficient conditions for global existence and blow up of the weak solutions in the supercritical state(J(u0,v0)>d)are given. |
