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## A COMPARISON THEOREM FOR NONLINEAR TIME-FRACTIONAL DIFFUSIONS

Let $u,v$ be mild solutions of $Du + Au = g(u)$ and $Dv + Av = h(v)$ where $g,h$ are Holder continuous of order $\beta\in (0,1)$, and $D = D^\beta_t$ is the Caputo fractional time derivative.  If $g(\zeta) \le h(\zeta)$ for all $\zeta \in \mathbf{R}$ and $u(\cdot,0) \le v(\cdot,0)$ then $u(x,t) \le v(x,t)$ for all $(x,t) \in \mathbf{R}\times [0,T]$.

This is an adaptation of Engler’s Proposition 4.1 (engler-spread-speed-reaction-diffusions),  Define $\hat{g}(\zeta) = e(\lambda,t) g( e^{-1}(\lambda, t) \zeta)$ and similarly for $\hat{h}$ where $e(\lambda,t) = E_{\beta}(\lambda t^\beta)$.  Define

$U(x,t) = e(\lambda,t) u(x,t)$

$V(x,t) = e(\lambda,t) v(x,t)$

Then $D U + AU = \hat{g}(t,U)$ and $D V + AV = \hat{h}(t,V)$.  By Lemma 2.6 of Dabas-Chauhan-Kumar (Dabas-Chauhan-Kumar-TimeFracSolution) assuming $U_n \le V_n$ we must prove $U_{n+1} \le V_{n+1}$.  First use $u(\cdot,0) \le v(\cdot,0)$ and $\hat{g}(s,U_n)\le \hat{g}(s,V_n)$ finally $\hat{g}(s,V_n) \le \hat{h}(s,V_n)$ using positivity of $T_{\beta}$ and $S_{\beta}(t)$

$U_{n+1} = T_{\beta}(t) u(\cdot,0) + \int_0^t S_{\beta}(t-s) \hat{g}(s,U_n)ds$

$\le T_{\beta}(t) v(\cdot,0) + \int_0^t S_{\beta}(t-s)\hat{h}(s,V_n)ds = V_{n+1}$

This proves the theorem.