Blow-up for the ω-heat equation with Dirichlet boundary conditions and a reaction term on graphs

摘要

In this paper, we consider the blow-up phenomenon for the ω-heat equation on graph with Dirichlet boundary conditions and a reaction term u_t (x, t) = △_ωu(x, t) + u~p(x, t), where △_ω is called the discrete weighted Laplacian operators. If p ≤ 1, every solution is global, and if p > 1 and under some suitable conditions, we prove that the nonnegative and nontrivial solution blows up in finite time and the blow-up rate on L~∞-norm only depends on p. Finally, two examples are proposed to demonstrate our results.