On diffusive population models with toxicants and time delays. (English) Zbl 0927.35049

The authors study two generalized reaction-diffusion systems, as logistic systems in mathematical ecology. The first of these is the following: \[ \partial u(t,x)/\partial t- Au(t,x)= r(x) u(t,x)(K(x)- u(t,x))/(K(x)+ c(x)u(t,x))\text{ in }[0,\infty)\times \Omega, \]
\[ B[u](t,x)= 0\quad\text{on }(0,\infty)\times \partial\Omega,\quad u(0,x)= u_0(x)\quad\text{on }\overline\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary \(\partial\Omega\); the functions \(r\), \(c\) and \(K\) are positive in \(\Omega\) and Hölder continuous on \(\overline\Omega\); \(B[u]= u\) or \(B[u]= \partial u/\partial\nu+ \gamma(x)u\), with \(\gamma\in C^{1+\alpha}(\partial\Omega)\) and \(\gamma(x)\geq 0\) on \(\partial\Omega\); the initial function \(u_0\) is a Hölder continuous function on \(\overline\Omega\), and the differential operator \(A\) is a uniformly strongly elliptic operator.
The main goal of this paper is to show the existence of a unique positive steady-state solution in these models and investigate the asymptotic behavior of the time-dependent solutions, in both models, in relation to such steady solutions.


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
92D40 Ecology
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