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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.

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
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