On a p-Laplace type evolution equation with random variable exponent and a stochastic force

我们对具有随机变量指数和随机力的p-拉普拉斯型非线性抛物方程感兴趣。由于随机变指数，该问题不适合经典Bochner空间的框架，将在概率空间上的变指数Lebesgue和Sobolev空间中进行研究。本文采用随机偏微分方程中已知的经典单调性方法和工具来证明Dirichlet问题弱解的存在唯一性。此外，我们对随机能量问题和所研究方程解的Itô公式的存在感兴趣。我们还想研究希尔伯特空间中有值的布朗运动的推广。

We are interested in a p-Laplace type nonlinear parabolic equation with a random variable exponent and a stochastic force. Due to the random variable exponent, this problem does not fit into the framework of classical Bochner spaces and will be studied in Lebesgue and Sobolev spaces with variable exponent on a probability space. We adapt classical monotonicity methods and tools known from stochastic partial differential equations to show existence and uniqueness of weak solutions of the Dirichlet problem. Moreover we are interested in the problem of stochastic energy and in the existence of an Itô Formula for the solutions of the studied equation. We also want to investigate generalizations for Brownian motion with values in Hilbert spaces.