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arXiv · 2207.14110

Well-posedness for Cauchy fractional problems involving discrete convolution operators

Jorge González-Camus
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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^β_t u(n,t)= B u(n,t) + f(n-ct,u(n,t)),\, &n\in\mathbb{Z}, \;t>0, u(n,0)=φ(n),\; &n\in\mathbb{Z}, \end{cases} \end{equation*} under the assumptions that $β\in (0,1]$, $c>0$ 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.

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Jorge González-Camus. 2022-08-16. Well-posedness for Cauchy fractional problems involving discrete convolution operators. https://arxiv.org/abs/2207.14110

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