The stochastic $p$-Laplace equation on $\mathbb{R}^d$
Abstract
We show well-posedness 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 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 $1<p<\infty$ 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 fixed-point argument yields the result for multiplicative noise.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2020
- DOI:
- 10.48550/arXiv.2012.10148
- arXiv:
- arXiv:2012.10148
- Bibcode:
- 2020arXiv201210148S
- Keywords:
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- Mathematics - Probability;
- Mathematics - Analysis of PDEs;
- 35K55;
- 35R60;
- 60H15