Well-posedness for Cauchy fractional problems involving discrete convolution operators

TL;DR

This paper establishes conditions for the well-posedness of a nonlinear fractional semidiscrete model involving discrete convolution operators, focusing on existence, uniqueness, and comparison principles of solutions.

Contribution

It introduces new sufficient conditions ensuring well-posedness for fractional problems with discrete convolution operators, including existence, uniqueness, and comparison results.

Findings

Proved existence and uniqueness of solutions.

Established a comparison principle for solutions.

Identified conditions on the convolution kernel and nonlinearity.

Abstract

This work is focused on establishing sufficient conditions to guarantee the well-posedness of the following nonlinear fractional semidiscrete model \begin{equation*} \begin{cases} \mathbb D^\beta_t u(n,t)= B u(n,t) + f(n-ct,u(n,t)),\, &n\in\mathbb{Z}, \;t>0, u(n,0)=\varphi(n),\; &n\in\mathbb{Z}, \end{cases} \end{equation*} under the assumptions that $\beta \in (0,1]$, $c>0$ some constant, $B$ is a discrete convolution operator with kernel $b\in\ell^1(\Z)$, which is the infinitesimal generator of the Markovian $C_0$-semigroup and suitable nonlinearity $f$. We present results concerning the existence and uniqueness of solution, as well as establishing a comparison principle of solutions according to respective initial values.