- Erkan Nane, Auburn University
- Blow-up Results for Space--time Fractional Dynamics
- 10/18/2018
- 3:00 PM - 3:50 PM
- C405 Wells Hall
Linked Abstract
ABSTRACT. Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type,
$$
\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[b(u)+ \sigma(u)\stackrel{\cdot}{F}(t,x)]
$$
in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$, $\alpha\in (0,2]$. The operator $\partial^\beta_t$ is the Caputo fractional derivative while $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic $\alpha$-stable L\'evy process and $I^{1-\beta}_t$ is the Riesz fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. These equations might be used as a model for materials with random thermal memory. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on $b$, $\sigma$ and the initial condition. Our results complement those of P. Chow in ``P.-L. Chow. Unbounded positive solutions of nonlinear parabolic It$\hat{o}$ equations. Commun. Stoch. Anal., 3(2)(2009), 211--222.'' and ``P.-L. Chow. Explosive solutions of stochastic reaction-diffusion equations in mean $l_{p}$-norm. J. Differential Equations, 250(5) (2011), 2567--2580.'' and Foondun and Parshad ``M. Foondun and R. Parshad, On non-existence of global solutions to a class of stochastic heat equations. Proc. Amer. Math. Soc. 143 (2015), no. 9, 4085--4094'', among others. The results presented are our recent joint work with Sunday Asogwa, Mohammud Foondun, Wei Liu, and Jebessa Mijena.