The supplementation of the theory of periodic solutions for a class of nonlinear diffusion equations

Abstract

In this paper, we provide supplementation of the theory of periodic solutions for a class of nonlinear diffusion equations ut−Δum=a(x,t)up with m > 0, p > 0. By using a new method to deal with the “blow up sequence”, we obtain the L∞ estimate with optimal exponent 1<pm<ps, where ps is the Sobolev critical exponent, then we apply topological degree method to prove the existence of positive periodic solutions. If pm≥ps,N≥3 and Ω is star–shaped, we establish Rellich–Pohozaev type identity to prove the nonexistence of positive periodic solutions. We improve the results of Mizoguchi [10, Therem 1] by filling the gap m>1,pm≥1+2mN. In addition, we also give quite complete results about the existence of periodic solutions for the fast diffusion case 0 < m < 1. As far as we know, there are no results about the fast diffusion case before present work.