arXiv subjects

Piotr Knosalla

Publications and source records attributed to Piotr Knosalla.

4 recordsLinked to original sources

arXiv2026

A chemotaxis-consumption model with boundary inflow of a nutrient, steady states

In this manuscript we consider the steady state problem for a chemotaxis-consumption model with positive inflow of a nutrient across the boundary. We show that in any bounded smooth domain there exists a unique nonconstant steady state provided that the mass of bacteria is sufficiently large. We further prove that, if the domain is a ball, this condition can be dropped and solutions exist for any positive bacterial mass.

math.AP↗

arXiv2025

The impact of Robin boundary condition on a chemotaxis-consumption-growth model

We investigate a parabolic-elliptic chemotaxis-consumption-growth system with a Robin boundary condition imposed on the signal. First, we analyse the steady state problem, then we show that the solutions of the system are global and uniformly bounded in time in any space dimension. Next, under smallness assumption on the boundary data, we show that the solutions converge to non-constant steady states as time tends to infinity.

math.AP↗

arXiv2024

Asymptotics of a chemotaxis-consumption-growth model with nonzero Dirichlet conditions

This paper concerns the asymptotics of certain parabolic-elliptic chemotaxis-consumption systems with logistic growth and constant concentration of chemoattractant on the boundary. First we prove that in two dimensional bounded domains there exists a unique global classical solution which is uniformly bounded in time, and then we show that if the concentration of chemoattractant on the boundary is sufficiently low then the solution converges to the positive steady state as time goes to infinity.

math.AP↗

arXiv2023

Self-similar solution for fractional Laplacian in cones

We construct a self-similar solution of the heat equation for the fractional Laplacian with Dirichlet boundary conditions in every fat cone. As applications, we give the Yaglom limit and entrance law for the corresponding killed isotropic stable Lévy process and precise large-time asymptotics for solutions of the Cauchy problem in the cone.

math.PR↗