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The existence of periodic solutions to reaction-diffusion systems with periodic data. (English) Zbl 0849.35052
The authors consider the parabolic system $$u_t= D(t) \Delta u+ f(x, t, u)\quad \text{in} \quad \Omega\times (0, \infty),$$ where $u= (u_1,\dots, u_m)$, $D= \text{diag}(d_i(t))$, $0< a\le d_i(t)\le b$, $\Omega\subset \bbfR^n$ bounded with smooth boundary, together with boundary conditions $u(x, t)= g(x, t)\ge 0$ on $\partial\Omega\times (0, \infty)$, $u(x, 0)= u_0(x)\ge 0$ in $\overline\Omega$. The main result states that this problem has a nonnegative $T$-periodic solution if $D(t)$ and $f(x, t, \xi)$ are $T$-periodic in $t$, $f$ is quasipositive ($f_i\ge 0$ if $\xi\ge 0$, $\xi_i= 0$) and growth conditions $|f(x, t, \xi)|\le K(1+ |\xi|^p)$, $\sum^i_{j= 1} \alpha_{ij} f_j(x, t, \xi)\le K(1+ \sum^m_1 \xi_j)$ for $\xi\ge 0$ with $\alpha_{ij}\ge 0$, $a_{ii}> 0$ ($i= 1,\dots, m$ with a special provision for $i= m$) hold. The proof uses a fixed point argument for the Poincaré map $u_0\mapsto u(., T)$. It carries over to Robin type boundary condition; in the Neumann case a more stringent growth condition on $f$ is required.

35K57Reaction-diffusion equations
35B10Periodic solutions of PDE
35K40Systems of second-order parabolic equations, general
Poincaré map
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