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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 $\bbfR^n$ 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 $\gamma(x)\ge 0$ 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.

MSC:
35K60Nonlinear initial value problems for linear parabolic equations
35K57Reaction-diffusion equations
35B40Asymptotic behavior of solutions of PDE
92D40Ecology
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