The stochastic p-Laplace equation on ℝ d

Abstract

We show well-posedness of the p-Laplace evolution equation on R d with square integrable random initial data for arbitrary 1 < p < ∞ and arbitrary space dimension d ∈ N . The noise term on the right-hand side of the equation may be additive or multiplicative. Due to a lack of coercivity of the p-Laplace operator in the whole space, the possibility to apply well-known existence and uniqueness theorems in the classical functional setting is limited to certain values of 1 < p < ∞ and also depends on the space dimension d. We propose a framework of functional spaces which is independent of Sobolev space embeddings and space dimension. For additive noise, we show existence using a time discretization. Then, a fixed-point argument yields the result for multiplicative noise.