The stochastic $p$Laplace equation on $\mathbb{R}^d$
Abstract
We show wellposedness of the $p$Laplace evolution equation on $\mathbb{R}^d$ with square integrable random initial data for arbitrary $1<p<\infty$ and arbitrary space dimension $d\in\mathbb{N}$. The noise term on the righthand 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 wellknown existence and uniqueness theorems in the classical functional setting is limited to certain values of $1<p<\infty$ 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 fixedpoint argument yields the result for multiplicative noise.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.10148
 Bibcode:
 2020arXiv201210148S
 Keywords:

 Mathematics  Probability;
 Mathematics  Analysis of PDEs;
 35K55;
 35R60;
 60H15