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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,
  • Gerardo Pérez-Suárez

摘要

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 \(\{\tau <\infty \}\) { τ < } . For the case where G(x) is of the form \(C|x|^{1+\beta }\) C | x | 1 + β for some constants \(\beta ,C>0\) 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.