Global behavior of nonlocal in time reaction-diffusion equations

Abstract

The present paper considers the Cauchy-Dirichlet problem for the time-nonlocal reaction-diffusion equation $${\partial }_t(k\ast (u-u_0))+{\mathcal{L}}_x[u]=f(u),x\in \Omega \subset {\mathbb{R}}^n,t>0,$$ where $k\in L_{loc}^1({\mathbb{R}}_{+}),$ $f$ is a locally Lipschitz function, ${\mathcal{L}}_x$ is a linear operator. This model arises when studying the processes of anomalous and ultraslow diffusions. Results regarding the local and global existence, decay estimates, and blow-up of solutions are obtained. The obtained results provide partial answers to some open questions posed by Gal and Varma (2020), as well as Luchko and Yamamoto (2016). Furthermore, possible quasi-linear extensions of the obtained results are discussed, and some open questions are presented.

16 pages

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• Trends, Directions for Further Research, and Some Open Problems of Fractional Calculus
• Sobolev, Hardy, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg type inequalities for some fractional derivatives