Quenching of the solution to the discrete heat equation with logarithmic type sources on graphs

ABSTRACT

In the current paper, we mainly consider the following discrete heat equation on graphs with logarithmic type sources: $\begin{array}{rl}& u_t={\mathrm{\Delta }}_{\rho }u+\lambda \ln u,x\in S,t\in (0,T_{max}),\\ & u(x,t)=1,x\in \mathrm{\partial }S,t\in (0,T_{max}),\\ & u(x,0)=u_0\left(x\right),x\in S.\end{array}$ First, the local existence and uniqueness of the above problem are obtained via the Banach fixed point theorem. By the comparison principle, the quenching behavior of the above problem and the blow-up of its time-derivatives of its solution at finite time under some suitable conditions are also given. Moreover, we also obtain the existence of the critical exponent ${\lambda }^{\ast }$, when $\lambda \le {\lambda }^{\ast }$, the above problem admits a global solution by the implicit function theorem. On the other hand, as $\lambda >{\lambda }^{\ast }$, its solution will quench at finite time. Finally, a numerical experiment on the graph with five vertexes is used to explain the theoretical results.

COMMUNICATED BY:

• Salvatore Leonardi

KEYWORDS:

• Discrete heat equation
• logarithmic type sources
• quenching
• blow-up
• graphs

AMS SUBJECT CLASSIFICATIONS:

• 35A01
• 35K55
• 35B40

Disclosure statement

No potential conflict of interest was reported by the authors.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work is mainly supported by Natural Science Foundation (NSF) of China (grant no. 11461075).