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

José Alfredo López-Mimbela () and Gerardo Pérez-Suárez ()
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José Alfredo López-Mimbela: Centro de Investigación en Matemáticas
Gerardo Pérez-Suárez: Centro de Investigación en Matemáticas

Methodology and Computing in Applied Probability, 2024, vol. 26, issue 4, 1-21

Abstract: Abstract 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.

Keywords: Explosion in finite time; Blowup probability; Blowup rate; Blowup time; SPDEs; Random PDEs; Feynman-Kac semigroup; Exponential functionals; 60H15; 35R60; 35B44; 60L20; 35K58; 35B40 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11009-024-10123-9

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