Abstract
In this paper, we discuss the conditions under which blow-up occurs for the solutions of reaction-diffusion equations on networks. The analysis of this class of problems includes the existence of blow-up in finite time and the determination of the blow-up time and the corresponding blow-up rate. In addition, when the solution blows up, we give estimates for the blow-up time and also provide the blow-up rate. Finally, we show some numerical illustrations which describe the main results.
Highlights
We say that a solution u to the equation blows up at time T, if |u(xn, tn)| → +∞ for some sequence → (a, T )
There have been many papers which study the blow-up phenomenon for the solution to the reaction-diffusion equations
From a similar point of view, it will be interesting to investigate the diffusion of energy or information on networks, which can be modelled by the discrete reaction-diffusion equation on networks
Summary
We say that a solution u to the equation blows up (or is a thermal runaway) at time T, if |u(xn, tn)| → +∞ for some sequence (xn, tn) → (a, T ). There have been many papers which study the blow-up phenomenon for the solution to the reaction-diffusion equations. Since functions on a finite network of size N can be identified with vectors in RN , the theoretical framework is the classical theory for systems of ODE It seems that blow-up solution or global solution can be obtained by virtue of the famous Wintner’s Lemma (see [10]). We organized this paper as follows: After considering some concepts on networks and the local existence of solutions to the equation (1), we discuss the comparison principles on networks in order to study the blow-up phenomenon, in which we find out blow-up conditions of the solution to the equation (1) and the blow-up time with the blow-up rate.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
