Estimates of solutions of impulsive parabolic equations and applications to the population dynamics
type: Item
instance of: scholarly article
title: Estimates of solutions of impulsive parabolic equations and applications to the population dynamics
zbMATH Open document ID: 0871.35018
DOI: 10.5565/PUBLMAT_40196_06
published in: Publicacions Matemàtiques
publication date: 1997-07-27T00:00:00Z
full work available at URL: https://eudml.org/doc/41252
review text: The following reaction-diffusion equation is considered \[ u_t(t,x)=\kappa\Delta u(t,x)+u(t,x) f_0(u(t,x))+c_0, \quad (t,x)\in\Gamma, \] subject to the initial condition \(u(0,x)=u_0(x),\quad x\in\bar\Omega,\) the boundary condition \(u(t,x)=0,\quad (t,x)\in\mathbb{R}_+\times\partial\Omega\) and the impulses at fixed moments \[ u(t_k,x)=u(t_k^-,x)+g(t_k,x,u(t_k^-,x)),\quad x\in\bar\Omega,\quad k=1,2,\ldots . \] Here the following notations are used: \(\Omega\subset \mathbb{R}^2\) is a bounded domain with smooth boundary \(\partial\Omega\) and \(\bar\Omega=\Omega\cup\partial\Omega\); \(0=t_0<t_1<t_2<\cdots<t_k<\cdots\) are given numbers with \(\lim_{k\to\infty}t_k=+\infty\), \(E=\{(t,x)\in\mathbb{R}^{1+n}\mid t\in \mathbb{R}_+,x\in\bar\Omega\}, \Gamma_k=\{(t,x)\in E\mid t\in(t_k,t_{k+1}), x\in\Omega\}\); \(\Gamma=\cup_{k=0}^\infty\Gamma_k\). This problem describes a single species population in a bounded environment. The function \(u(t,x)\) represents the population density at the point \(x\in\bar\Omega\) and time \(t\geq0\), \(f_0:\mathbb{R}\to\mathbb{R}\) is the specific growth rate of \(u\), \(\kappa>0\) is the diffusion coefficients, \(c_0\geq0\) is a constant. The function \(g:\{t_k\}_{k=1}^\infty\times\bar\Omega\times \mathbb{R}\to\mathbb{R}\) describes instantaneous changes in the population density due to phenomena as harvesting, disaster, immigration, emigration, etc. In the paper the growth of the solutions of the impulsive parabolic equation is established by means of solutions of an ordinary differential equation.
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34A37
zbMATH DE Number: 911927
zbMATH Keywords: growth rate
author: Drumi Bainov
author: Emil Minchev
MaRDI profile type: Publication
OpenAlex ID: W2097386766
description: scientific article; zbMATH DE number 911927