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

Dario D. Monticelli; Fabio Punzo; Jacopo Somaglia

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

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arxiv 2404.12058 v3 pith:YCENENUY submitted 2024-04-18 math.AP

Dario D. Monticelli , Fabio Punzo , Jacopo Somaglia This is my paper

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keywords solutionsweightedgraphsnonexistenceparabolicpotentialassumingbound

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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.

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Nonexistence results for the semilinear wave equation on graphs
math.AP 2025-06 reject novelty 6.0 of 10

On infinite weighted graphs, if the graph Laplacian of the distance grows slowly and suitable weighted volume growth holds, every very weak solution of the semilinear wave inequality is identically zero.

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