Blowup Probability, Blowup Time and Blowup Rate of Nonlinear Heat Equations with Potential Term Perturbed by a Multiplicative Gaussian Rough Noise

In this article we investigate the blowup behavior of semilinear stochastic partial differential equations of the prototype $$\begin{aligned} \textrm{d}u(t,x)=\left[ -(-\Delta )^{\alpha /2}u(t,x)-q(x)u(t,x) +\gamma u(t,x)+G(u(t,x))\right] \,\textrm{d}t+\kappa u(t,x)\,\textrm{d}Z_t \end{aligned}$$ d u ( t , x ) = - ( - Δ ) α / 2 u ( t , x ) - q ( x ) u ( t , x ) + γ u ( t , x ) + G ( u ( t , x ) ) d t + κ u ( t , x ) d Z t on the space $$\mathbb {R}^{d}$$ R d , where $$\alpha \in (0,2]$$ α ∈ ( 0 , 2 ] , $$\gamma ,\kappa \in \mathbb {R}$$ γ , κ ∈ R , q is a nonnegative locally bounded function such that $$q(x)\rightarrow \infty $$ q ( x ) → ∞ as $$|x|\rightarrow \infty $$ | x | → ∞ , G is a locally Lipschitz continuous function, and Z is a real-valued centered Gaussian process with Hölder continuous paths with exponent $$\theta >1/3$$ θ > 1 / 3 . We show that there exists a random maximal interval $$[0,\tau )$$ [ 0 , τ ) where the solution exists and is unique. Moreover, we show that the solution explodes in $$L^{\infty }$$ L ∞ -norm on the event $$\{\tau 0$$ β , C > 0 , we obtain estimates for the blowup probability, the blowup time, and the blowup rate of the solution. These bounds are given in terms of the exponential functional of Z.