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, - Δ p u , can have a singular part in accordance with the Lebesgue decomposition theorem. The singular portion of - Δ p u corresponds to where - Δ p u = - ∞ , 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 - Δ p u despite a complete lack of estimates in the time direction. However, we must assume that 1 < p ≤ 2 .