Gaussian estimates for a heat equation on a network

TL;DR

This paper establishes Gaussian estimates for a diffusion equation on a network with complex boundary conditions, demonstrating well-posedness, semigroup properties, and applications to neurobiological models.

Contribution

It provides the first Gaussian upper bounds for heat kernels on networks with non-local Kirchhoff conditions, extending semigroup theory to these settings.

Findings

Proved well-posedness of the diffusion problem on networks.

Derived Gaussian upper bounds for the integral kernel.

Applied results to semilinear systems in neurobiology.

Abstract

We consider a diffusion problem on a network on whose nodes we impose Dirichlet and generalized, non-local Kirchhoff-type conditions. We prove well-posedness of the associated initial value problem, and we exploit the theory of sub-Markovian and ultracontractive semigroups in order to obtain upper Gaussian estimates for the integral kernel. We conclude that the same diffusion problem is governed by an analytic semigroup acting on all $L^p$-type spaces as well as on suitable spaces of continuous functions. Stability and spectral issues are also discussed. As an application we discuss a system of semilinear equations on a network related to potential transmission problems arising in neurobiology.