Narrow Results

In this paper, a differential-geometric method is applied to build some Li–Yau–Hamilton-type Harnack inequalities for the positive solutions to a one spatial dimensional nonlinear reaction–diffusion equation in a plane geometry. The class of reaction–diffusion equation that is considered here contains several important equations some of which are Newel–Whitehead–Segel, Allen–Cahn and Fisher–KPP equations. The Harnack inequalities so derived are used to discuss some other important properties of positive solutions and in the characterization of positive wave solutions.

There is a huge literature about Harnack inequalities so here I restrict my talk to the case of weak solutions of elliptic equations in divergence form mentioning only fundamental contributions obtained by means of purely analytic tools and giving few historical references. I focus my attention on Harnack inequalities for either Dirichlet forms or p-homogeneous energy forms on fractal sets.