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Propagation and blocking in periodically hostile environments. (English) Zbl 1263.35036
The main purpose of this paper is to study the persistence and propagation phenomena for one species in a periodically hostile environment. This is modeled by the following reaction-diffusion equation $\begin{cases} u_t-\nabla\cdot (A(x)\nabla u)=F(x,u), & \;x\in \overline\Omega \\ u(t,x)=0, & \;x \in \partial \Omega \\ \end{cases}$ where $$\Omega$$ is an unbounded domain in $$\mathbb{R}^N$$, $$A(x)=(A_{ij}(x))$$ is a symmetric matrix and is uniformly positive definite, $$F(x,u)$$ is assumed to satisfy the condition that for all $$x\in \overline\Omega$$, $$F(x,0)=0$$ and $$F(x,M)\leq 0$$ for some $$M>0$$. In addition, the domain $$\Omega$$, the functions $$A_{ij}(\cdot)$$, and the function $$F(\cdot,u)$$ are all periodic in the sense that they are invariant under the translation $$x\rightarrow x+k$$ for $$k\in L_1\mathbb{Z}\times \cdots \times L_N\mathbb{Z}$$, where $$L_1, \cdots, L_N$$ are fixed positive numbers. A result on the existence of a minimal nonnegative stationary solution and the asymptotic behavior of the solution of the initial boundary value problem is presented.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K57 Reaction-diffusion equations 35K58 Semilinear parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations
##### Keywords:
minimal nonnegative stationary solution
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