On the critical behavior for time-fractional reaction diffusion problems

Abstract

We first investigate the existence and nonexistence of weak solutions to the time-fractional reaction diffusion problem \begin{document}$ \frac{\partial^\alpha u}{\partial t^\alpha}-\frac{\partial^2 u}{\partial x^2}+u\geq x^{-a}|u|^p, \, \, t>0, \, x\in (0, 1], \quad u(0, x) = u_0(x), \, \, x\in (0, 1] $\end{document} under the inhomogeneous Dirichlet boundary condition \begin{document}$ u(t, 1) = \delta, \quad t>0, $\end{document} where $ u = u(t, x) $, $ 0<\alpha<1 $, $ \frac{\partial^\alpha }{\partial t^\alpha} $ is the time-Caputo fractional derivative of order $ \alpha $, $ a\geq 0 $, $ p>1 $ and $ \delta>0 $. We show that, if $ a\leq 2 $, the existence holds for all $ p>1 $ while if $ a>2 $, then the dividing line with respect to existence or nonexistence is given by the critical exponent $ p^* = a-1 $. The proof of the nonexistence result is based on nonlinear capacity estimates specifically adapted to the nonlocal nature of the problem, the modified Helmholtz operator $ -\frac{\partial^2}{\partial x^2}+I $, and the considered boundary condition. The existence part is proved by the construction of explicit solutions. We next extend our study to the case of systems.