Nonexistence of solutions to parabolic problems with a potential on weighted graphs

TL;DR

This paper proves that under certain conditions, no nontrivial nonnegative solutions exist for specific semilinear parabolic equations on weighted graphs, highlighting a critical exponent phenomenon similar to Fujita's results.

Contribution

It establishes nonexistence results for solutions on weighted graphs under volume growth and Laplacian bounds, extending classical PDE results to graph settings.

Findings

No global solutions exist under specified conditions

Identifies critical exponent phenomenon on weighted graphs

Shows optimality of hypotheses for nonexistence

Abstract

We investigate nonexistence of nontrivial nonnegative solutions to a class of semilinear parabolic equations with a positive potential, posed on weighted graphs. Assuming an upper bound on the Laplacian of the distance and a suitable weighted space-time volume growth condition, we show that no global solutions exists. We also discuss the optimality of the hypotheses, thus recovering a critical exponent phenomenon of Fujita type.