A New Condition for Blow-up Solutions to Discrete Semilinear Heat Equations on Networks

Abstract

The purpose of this paper is to introduce a new condition \[ \hbox{(C)$\alpha {\int }_0^{u}f(s)ds\le uf(u)+\beta u^2+\gamma ,u>0$} \] for some $\alpha ,\beta ,\gamma >0$ with $0<\beta \le \frac{\left(\alpha -2\right){\lambda }_0}{2}$, where ${\lambda }_0$ is the first eigenvalue of discrete Laplacian ${\Delta }_{\omega }$, with which we obtain blow-up solutions to discrete semilinear heat equations \begin{equation*} \begin{cases} u_{t}\left(x,t\right)=Δ_ωu\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in S\times\left(0,+\infty\right),\\ u\left(x,t\right)=0, & \left(x,t\right)\in\partial S\times\left[0,+\infty\right),\\ u\left(x,0\right)=u_{0}\geq0(nontrivial), & x\in\overline{S} \end{cases} \end{equation*} on a discrete network $S$. In fact, it will be seen that the condition (C) improves the conditions known so far.

19 pages