An approximation of solutions to heat equations defined by generalized measure theoretic laplacians

Author: 

Hambly, B
Ehnes, T

Publication Date: 

19 August 2020

Journal: 

Journal of Evolution Equations

Last Updated: 

2021-04-06T01:44:41.177+01:00

DOI: 

10.1007/s00028-020-00602-0

abstract: 

We consider the heat equation defined by a generalized measure theoretic Laplacian on [0, 1]. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure μ, where we do not assume the existence of a strictly positive mass density. We show that weak measure convergence implies convergence of the corresponding generalized Laplacians in the strong resolvent sense. We prove that strong semigroup convergence with respect to the uniform norm follows, which implies uniform convergence of solutions to the corresponding heat equations. This provides, for example, an interpretation for the mathematical model of heat diffusion on a bar with gaps in that the solution to the corresponding heat equation behaves approximately like the heat flow on a bar with sufficiently small mass on these gaps.

Symplectic id: 

1130227

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article