Estimates of solutions of impulsive parabolic equations and applications to the population dynamics
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.
scientific article; zbMATH DE number 911927