The existence and nonexistence of global solutions for a semilinear heat equation on graphs

The existence and nonexistence of global solutions for a semilinear heat equation on graphs Let $$G=(V,E)$$ G = ( V , E ) be a finite or locally finite connected weighted graph, $$\Delta $$ Δ 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 $$\begin{aligned} \left\{ \begin{array}{lc} u_t=\Delta u + u^{1+\alpha } &{}\, \text {in }(0,+\infty )\times V,\\ u(0,x)=a(x) &{}\, \text {in }V. \end{array} \right. \end{aligned}$$ u t = Δ u + u 1 + α in ( 0 , + ∞ ) × V , u ( 0 , x ) = a ( x ) in V . We conclude that, for a graph satisfying curvature dimension condition $$\textit{CDE}'(n,0)$$ CDE ′ ( n , 0 ) and $$V(x,r)\simeq r^m$$ V ( x , r ) ≃ r m , if $$0<m\alpha <2$$ 0 < m α < 2 , then the non-negative solution u is not global, and if $$m\alpha >2$$ m α > 2 , 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 $${\mathbb {Z}}^m$$ Z m . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

The existence and nonexistence of global solutions for a semilinear heat equation on graphs

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1204-y
Publisher site
See Article on Publisher Site

Abstract

Let $$G=(V,E)$$ G = ( V , E ) be a finite or locally finite connected weighted graph, $$\Delta $$ Δ 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 $$\begin{aligned} \left\{ \begin{array}{lc} u_t=\Delta u + u^{1+\alpha } &{}\, \text {in }(0,+\infty )\times V,\\ u(0,x)=a(x) &{}\, \text {in }V. \end{array} \right. \end{aligned}$$ u t = Δ u + u 1 + α in ( 0 , + ∞ ) × V , u ( 0 , x ) = a ( x ) in V . We conclude that, for a graph satisfying curvature dimension condition $$\textit{CDE}'(n,0)$$ CDE ′ ( n , 0 ) and $$V(x,r)\simeq r^m$$ V ( x , r ) ≃ r m , if $$0<m\alpha <2$$ 0 < m α < 2 , then the non-negative solution u is not global, and if $$m\alpha >2$$ m α > 2 , 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 $${\mathbb {Z}}^m$$ Z m .

Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jul 7, 2017

References

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