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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 $$\mathbb{R}^n$$ 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.

MSC:
 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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References:
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