Nonexistence of solutions to parabolic problems with a potential on weighted graphs
Dario D. Monticelli, Fabio Punzo, Jacopo Somaglia
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 exist. We also discuss the optimality of the hypotheses, thus recovering a critical exponent phenomenon of Fujita type.
Topics & Concepts
MathematicsClass (philosophy)Upper and lower boundsFujita scaleLaplace operatorExponentParabolic partial differential equationApplied mathematicsMathematical analysisPure mathematicsp-LaplacianCombinatoricsHeat equationCritical exponentFractional LaplacianVolume (thermodynamics)Nonlinear Partial Differential EquationsAdvanced Mathematical Physics ProblemsDifferential Equations and Boundary Problems