Extinction and asymptotic behavior of solutions for the ω-heat equation on graphs with source and interior absorption

In this paper, we consider the extinction and asymptotic behavior of the solutions for the following ω-heat equation on graphs with source and interior absorption:ut(x,t)=Δωu(x,t)+λuq(x,t)−up(x,t),ut(x,t)=Δωu(x,t)+λuq(x,t)−up(x,t), where ΔωΔω is called the discrete weighted Laplacian operator and λ,p,q>0λ,p,q>0. We first prove the local existence of the solutions and show that all solutions exist globally when q≤max⁡{p,1}q≤max⁡{p,1}. Then, we obtain the following extinction and asymptotic properties of the solutions: when q>1q>1 and the initial datum are small enough, the solution vanishes in infinite time; when q=1q=1, p<1p<1 and λ   is appropriately small, the solution becomes extinct in finite time; when q=1q=1, p≥1p≥1 and λ   is appropriately small, the solution vanishes in infinite time; when q=1q=1 and λ   is appropriately large, the solution is non-extinction; when q<1q<1, q=pq=p is the critical extinction exponent. Moreover, for the case that p>q=1p>q=1 and λ   is appropriately large, we show that the non-extinction solution is asymptotically stable and limt→∞⁡u(x,t)=u+(x), where u+(x)u+(x) is the unique positive equilibrium solution. Finally, we demonstrate our results through some numerical examples.