A nonlinear wave equation with fractional perturbation

Abstract

We study a $d$-dimensional wave equation model ($2\leq d\leq4$) with quadratic nonlinearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter $H=(H_{0},\ldots,H_{d})\in(0,1)^{d+1}$ of the noise: If $\sum_{i=0}^{d}H_{i}>d-\frac{1}{2}$, then the equation can be treated directly, while in the case $d-\frac{3}{4}<\sum_{i=0}^{d}H_{i}\leq d-\frac{1}{2}$, the model must be interpreted in the Wick sense, through a renormalization procedure.

Our arguments essentially rely on a fractional extension of the considerations of [Trans. Amer. Math. Soc. 370 (2017) 7335–7359] for the two-dimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.