Parabolic theory of the discrete \(p\)-Laplace operator
nonlinear \(C_0\)-semigroups generated by subdifferentials
discrete nonlinear diffusion-type problem
fractional discrete differential equation
The structure of this paper is clear, and the process of proofs is strict. The words of the paper are appropriate and normative. This paper is well written and contains some interesting and correct results.NEWLINENEWLINEThe author studies the elementary features of the discrete \(p\)-heat equation NEWLINE\[NEWLINE\frac{d}{dt}\sum\limits_{v\in V}\varphi(t,v)=-\sum\limits_{v\in V} \mathcal{I}(|\mathcal{I}^{T}\varphi|^{p-2}{I}^{T}\varphi)(t,v) +\sum\limits_{v\in V}f(t,v).NEWLINE\]NEWLINENEWLINENEWLINEBy introducing the relevant functional setting, the author formulates the main well-posedness result for the discrete \(p\)-heat equation. He shows that the nonlinear \(C_0\)-semigroup generated by the discrete \(p\)-Laplacian consists of an irreducible, sub-Markovian operator. Also, by energy methods, he discusses how these \(C_0\)-semigroups interact with the symmetries of the graph, based on the notion of an almost equitable partition of a graph. At last, he overviews a few popular generalizations of the discrete Laplacian, proposes some more and suggests how the variational structure of the \(p\)-heat equation can be generalized to discuss a broad class of discrete nonlinear diffusion-type problems, including a discretized porous medium equation.NEWLINENEWLINEThe author obtains that the results follow from the standard theory of ordinary differential equations whenever \(G\) is a finite graph. And the results are more general. The results are new in the case of fractional discrete differential equations, so one can apply these results rather than those presented in this paper.
scientific article; zbMATH DE number 6246145
scientific article