Study of a fractional stochastic heat equation

In this article, we study a $d$-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise: \begin{equation*} \left\{\begin{array}{l} \partial_t u-\Delta u= \rho^2 u^2 + \dot B \, , \quad t\in [0,T] \, , \, x\in \mathbb{R}^d \, ,\\ u_0=\phi\, . \end{array} \right. \end{equation*} Two types of regimes are exhibited, depending on the ranges of the Hurst index $H=(H_0,...,H_d)$ $\in (0,1)^{d+1}$. In particular, we show that the local well-posedness of (SNLH) resulting from the Da Prato-Debussche trick, is easily obtained when $2 H_0+\sum_{i=1}^{d}H_i >d$. On the contrary, (SNLH) is much more difficult to handle when $2H_0+\sum_{i=1}^{d}H_i \leq d$. In this case, the model has to be interpreted in the Wick sense, thanks to a time-dependent renormalization. Helped with the regularising effect of the heat semigroup, we establish local well-posedness results for (SNLH) for all dimension $d\geq1.$