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Authors: Fila, Marek | Winkler, Michael

Article Type: Research Article

Abstract: We analyze a mathematical model introduced by Anguige, Ward and King (J. Math. Biol. 51 (2005), 557–594) to describe a quorum sensing mechanism in spatially-structured populations of the bacteria Pseudomonas aeruginosa. In the biologically relevant limit case when the spatial distribution of bacteria is constant in time, this model reduces to a single semilinear parabolic equation for the concentration of the signal substance N-(3-oxododecanoyl)-homoserine lactone (AHL). We show that under some mild technical assumptions on the nonlinear AHL production rate, the AHL concentration approaches a uniquely determined steady state, and that this convergence takes place at an exponential rate.

Keywords: semilinear parabolic equation, quorum sensing, steady state, stabilization

DOI: 10.3233/ASY-2008-0920

Citation: Asymptotic Analysis, vol. 62, no. 1-2, pp. 89-106, 2009

Authors: Fila, Marek | Ishige, Kazuhiro | Kawakami, Tatsuki

Article Type: Research Article

Abstract: We consider the following initial value problem for a two-dimensional semilinear elliptic equation with a dynamical boundary condition: −Δu=up , x∈R2 + , t>0, ∂t u+∂ν u=0, x∈∂R2 + , t>0, u(x,0)=φ(x1 )≥0, x=(x1 ,0)∈∂R2 + , where u=u(x,t), ∂t :=∂/∂t, ∂ν :=−∂/∂x2 , R2 + :={(x1 ,x2 ): x1 ∈R,x2 >0} and p>1. We show that small solutions behave asymptotically like suitable multiples of the Poisson kernel. This is an extension of previous results of the authors of this paper to the two-dimensional case.

Keywords: semilinear elliptic equation in a half-plane, dynamical boundary condition, large time behavior

DOI: 10.3233/ASY-131183

Citation: Asymptotic Analysis, vol. 85, no. 1-2, pp. 107-123, 2013

Authors: Fila, Marek | Ishige, Kazuhiro | Kawakami, Tatsuki | Lankeit, Johannes

Article Type: Research Article

Abstract: We study the heat equation on a half-space or on an exterior domain with a linear dynamical boundary condition. Our main aim is to establish the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.

Keywords: Heat equation, dynamical boundary condition, large diffusion limit

DOI: 10.3233/ASY-181517

Citation: Asymptotic Analysis, vol. 114, no. 1-2, pp. 37-57, 2019