Existence, Uniqueness And Long Time Behaviour Of The Positive Weak Solutions Of A Nonlinear Reaction-diffusion Equation With Singular Term

In this doctoral dissertation, we mainly consider the existence, uniqueness and long time behaviour of the positive weak solutions of a nonlinear reaction-diffusion equation with singular term. We will treat this equation on a bounded domainΩ(?)RN, imposing Dirichlet boundary conditions u│(?)Ω=0 and "suffi-cient smoothness" ofΩ.We mainly consider the existence, uniqueness and long time behaviour of the positive weak solutions of the following nonlinear reaction-diffusion equation with singular term:We firstly construct the subsolution and supersolution by the eigenfunction corresponding to the first eigenvalue of the Laplace operator. Then we can obtain the integrability of the singular term based on the integrability of the first eigenfunction. Then we obtain the definition of the weak solution of the nonlinear reaction-diffusion equation with singular term.Moreover, using the convergence of the weak solution of the approximate equation, we prove the existence of the nonlinear reaction-diffusion equation with singular term. More precisely, we introduce the comparison principle of the nonlinear reaction-diffusion equation with singular term in the weak sense. Then by constructing the positive subsolution, we prove that this solution must be positive if the singular equation exist solution. Then we obtain an approx-imate equation without singular term. Finally, using convergence theorem, we prove existence of the weak solution of the singular equation based on the con-vergence of the weak solution of the approximate equation and the convergence of every term in the approximate equation.Finally, we study the existence of a global attractor for the nonlinear reaction-diffusion equation with singular term. It is well known that in many cases the long-time behaviors of dynamical system, generated by evolutionary equations of the corresponding semigroups. We respectively prove the existence of the absorbing set in L2 and H01∩LP, and the existence of the global attractor in Lp.