On Nonlinear Nonlocal Diffusion Equations

This is a study of a class of nonlocal nonlinear diffusion equations (NNDEs). We present several new qualitative results for nonlocal Dirichlet problems. It is shown that solutions with positive initial data remain positive through time, even for nonlinear problems; in addition, we prove that solutions to these equations obey a strong maximum principle. A striking result shows that nonlocal solutions must have some irregularity at the boundary; otherwise, we have ill-posedness of the initial value problem. In addition, for several general classes of NNDEs and nonlocal reaction-diffusion equations, we obtain explicit differential inequalities that bound above and below the solutions’ energy decay. Explicit solutions to these inequalities (hence, explicit rates of decay) are found for some special cases. Finally, we present a proof of the nonlocal Poincaré inequality with an explicit constant.