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In this paper, we deal with the blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphs. Under appropriate conditions, we prove that the nonnegative solution of the parabolic system blows up in a finite time on finite graphs and locally finite graphs, respectively. The results obtained extend some previous results of [Y. Lin and Y. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci. Ser. B 38(3) (2018) 843–856; Y. Wu, Local existence and blow-up of solutions to Fujita-type equations involving general absorption term on finite graphs, Fractals 30(2) (2022) 2240053]. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a simple, finite, connected, weighted graph satisfying curvature condition C D E ′ ( n , 0 ) and polynomial volume growth V ( x , r ) ≤ c 0 r m , Δ η be the normalized Laplacian. In this paper we prove that the semilinear heat equation u t = Δ η u + u 1 + α on G has no non-negative global solutions for any bounded, non-negative and non-trivial initial value in the case of m α = 2 . The obtained result provides a significant complement to the work that was done recently by Lin and Wu (2017) concerning the existence and nonexistence of global solutions for the semilinear heat equation on graphs. [ABSTRACT FROM AUTHOR]