Search

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]

Let $$G=(V,E)$$ be a finite or locally finite connected weighted graph, $$\Delta $$ be the usual graph Laplacian. Using heat kernel estimates, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on G We conclude that, for a graph satisfying curvature dimension condition $$\textit{CDE}'(n,0)$$ and $$V(x,r)\simeq r^m$$ , if $$02$$ , then there is a non-negative global solution u provided that the initial value is small enough. In particular, these results apply to the lattice $${\mathbb {Z}}^m$$ . [ABSTRACT FROM AUTHOR]