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

date: 7/17/2025, time: 16:00-17:30, room: ICS 118

Organizer: Ekaterina Kolkovska (CIMAT, Mexico)

Chair: Ekaterina Kolkovska (CIMAT, Mexico)

Explosion in finite time of solutions of a time-fractional semilinear heat equation

Gerardo Pérez-Suárez (Nanyang Technological University)

Time-fractional PDEs are useful models to describe anomalous diffusion. In this talk, we will consider a time-fractional Fujita equation obtained by replacing the classical derivative by the Caputo derivative. We will study the explosive behavior of its solution following a probabilistic approach. First, we will prove a stochastic representation for the solution in terms of a spatial branching process, which in some sense extends the classical branching Brownian motion to the fractional setting. Then, by using estimates for certain functionals of this process, we will provide estimates for the explosion time of the solution. This talk is based on ongoing joint work with N. Privault.

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

Ekaterina Todorova Kolkovska (Centro de Investigacion en Matematicas)

We study the blowup behaviour of a semilinear partial differential equation that is driven by a mixture of Brownian and fractional Brownian motion, on a bounded Lipschitz domain. The linear operator is supposed to have a strictly positive eigenfunction, and we show its influence, as well the influence of the other parameters of the equation, on the occurrence of a blowup in finite time. We give estimates for the probability of finite time blowup and of blowup before a given fixed time

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

José Alfredo López-Mimbela (Centro de Investigación en Matemáticas, Guanajuato, Mexico)

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 \(T\to\infty\).