Exponential crystal relaxation model with p-Laplacian

Abstract

In this article, we prove the global existence of weak solutions to an initial boundary value problem with an exponential and p-Laplacian nonlinearity. The equation is a continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation. In our investigation, we find a weak solution where the exponent in the equation, $-\Delta _p u$, can have a singular part in accordance with the Lebesgue decomposition theorem. The singular portion of $-\Delta _p u$ corresponds to where $-\Delta _p u = -\infty$, which leads it to have a canceling effect with the exponential nonlinearity. This effect has already been demonstrated for the case of a linear exponent $p=2$, and for the time-independent problem. Our investigation reveals that we can exploit this same effect in the time-dependent case with nonlinear exponent. We obtain a solution by first forming a sequence of approximate solutions and then passing to the limit. The key to our existence result lies in the observation that one can still obtain the precompactness of the term $e^{-\Delta _p u}$ despite a complete lack of estimates in the time direction. However, we must assume that $1<p\le 2$.