Function Spaces with a Random Variable Exponent

Abstract

The spaces with a random variable exponent $L^{p\left(\omega \right)}(D\times \Omega )$and $W^{k,p\left(\omega \right)}(D\times \Omega )$ are introduced. After discussing the properties of the spaces $L^{p\left(\omega \right)}(D\times \Omega )$ and $W^{k,p\left(\omega \right)}(D\times \Omega )$, we give an application of these spaces to the stochastic partial differential equations with random variable growth.