A Class Of Quasi-linear Parabolic Equations With Periodic Sources

In this paper, we mainly study the problem of non-trivial, non-negative periodic solutions of a class of quasi-linear parabolic equations with periodic sources under the homogeneous Dirichlet boundary condition, which contains the existence of periodic solutions, priori estimates, attractivity of maximal periodic solutions.In recent years, many authors have studied the problems of linear equations with nonlocal terms intensivly. At the same time, given the background of reality into account, many authors pay more attention to the problems of nonlinear periodic equations with nonlocal terms. So many authors pay more attention to nonlinear pericodic equations,especially to the differential equations with periodic sources, is mainly because that it has close relationship with other subject. The applications of the equation derived from physical, chemical, biological etc. So it has a very wide range of practical applications. The skill used for the semi-linear problem depended largely on the associated linearized periodic eigenvalue problem. Some authors studied the quasi-linear parabolic equations of divergence form, However, the method used most is not intuitive, and the result is not very perfect. At the same time, the study and use of the periodic parabolic eigenvalue problem with the divergence operator.In the study of this paper, we start from the basic theory of Sobolev space and periodic parabolic eigenvalue problem, taking the method of upper and lower solutions, comparison principle, Moser iteration embedding theorem, monotone iterative method, and some basic inequality as a tool to study the problem of non-trivial, non-negative periodic solutions of a class of quasi-linear parabolic equations with periodic sources.The methods we used in this paper is mainly the method of upper and lower solutions. Though the construction of upper and lower solutions, we establish the basement of the method of upper and lower solutions; Though comparison principle, monotone iterative method and the construction of Poincarémap, we solve the problem of the existence of periodic solutions; Moreover, though comparison principle Moser iteration, embedding theorem and some basic inequality, we make the priori estimates; At last, by using monotone iterative method, we give the asymptotic behavior of solutions.