CS27: Global and Non-Global Solutions of Semilinear Fractional Differential Equations

Organizer: Ekaterina Kolkovska (CIMAT)

Global and Non-global Solutions of a Fractional Reaction-Diffusion Equation Perturbed by a Fractional Noise

José Alfredo López-Mimbela

We provide conditions implying finite-time blowup of positive weak solutions to the semilinear equation \(d u(t,x) = \left[\Delta_{\alpha}u(t,x)+Ku(t,x) + u^{1+\beta}(t,x)\right]dt + \mu u(t,x)\, dB^H_t\), \(u(0,x)=f(x)\), \(x\in% \mathbb{R}^{\,d}\), \(t\ge0\), where \(\alpha\in(0,2]\), \(K\in\mathbb{R}\), \(% \beta>0\), \(\mu\ge0\) and \(H\in[\frac{1}{2},1)\) are constants, \(% \Delta_{\alpha}\) is the fractional power \(-(-\Delta)^{\alpha/2}\) of the Laplacian, \((B^H_t)\) is a fractional Brownian motion with Hurst parameter \(H\) , and \(f\ge0\) is a bounded measurable function. To achieve this we investigate the growth of integrals of the form \(\int^T \frac{e^{\beta (K s +\mu B^H_s)}}{s^{d\beta/\alpha}}\,ds\) as \(T\to\infty\). Moreover, we provide sufficient conditions for the existence of a global weak solution of the above equation, as well as upper and lower bounds for the probability that the solution does not blow up in finite time.

On the explosion time of a semilinear stochastic partial differential equations driven by a mixture of Brownian and fractional Brownian motion

Ekaterina Todorova Kolkovska

We study the blowup behaviour of a semilinear stochastic patial differential equation driven by a mixture of Brownian and fractional Brownian motion. We give estimates for the probability of finite time blowup and the blowup before a given fixed time. We show the influence of the biggest eigenvalue of the generator and the other parameters of the equation on the occurence of a blowup in finite time.