Exponential Crystal Relaxation Model With P-Laplacian

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 $1<p\leq 2$.