Blow-Up Solutions and Global Solutions to Discrete p-Laplacian Parabolic Equations

Abstract

We discuss the conditions under which blow-up occurs for the solutions of discrete $p$-Laplacian parabolic equations on networks $S$ with boundary $\partial S$ as follows: $u_t(x,t)={\mathrm{\Delta }}_{p,\omega }u(x,t)+\lambda |u(x,t){|}^{q-1}u(x,t)$, $(x,t)\in S\times (0,+\mathrm{\infty })$; $u(x,t)=0$, $(x,t)\in \partial S\times (0,+\mathrm{\infty })$; $u(x,0)=u_0\ge 0,$ $x\in \overline{S}$, where $p>1$, $q>0$, $\lambda >0$, and the initial data $u_0$ is nontrivial on $S$. The main theorem states that the solution $u$ to the above equation satisfies the following: (i) if and $q>1$, then the solution blows up in a finite time, provided ${\overline{u}}_0>{\left({\omega }_0/\lambda \right)}^{1/\left(q-p+1\right)}$, where ${\omega }_0:={\max }_{x\in S}{\sum }_{y\in \overline{S}}\mathrm{‍}\omega (x,y)$ and ${\overline{u}}_0:={\text{max}}_{x\in S}{\mathrm{\hspace{0.17em}u}}_0(x)$; (ii) if , then the nonnegative solution is global; (iii) if , then the solution is global. In order to prove the main theorem, we first derive the comparison principles for the solution of the equation above, which play an important role throughout this paper. Moreover, when the solution blows up, we give an estimate for the blow-up time and also provide the blow-up rate. Finally, we give some numerical illustrations which exploit the main results.