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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)Δu+f(x,t,u)inΩ×(0,),

where u=(u 1 ,,u m ), D=diag(d i (t)), 0<ad i (t)b, Ω n bounded with smooth boundary, together with boundary conditions u(x,t)=g(x,t)0 on Ω×(0,), u(x,0)=u 0 (x)0 in Ω ¯. The main result states that this problem has a nonnegative T-periodic solution if D(t) and f(x,t,ξ) are T-periodic in t, f is quasipositive (f i 0 if ξ0, ξ i =0) and growth conditions |f(x,t,ξ)|K(1+|ξ| p ), j=1 i α ij f j (x,t,ξ)K(1+ 1 m ξ j ) for ξ0 with α ij 0, a ii >0 (i=1,,m with a special provision for i=m) hold.

The proof uses a fixed point argument for the Poincaré map u 0 u(·,T). It carries over to Robin type boundary condition; in the Neumann case a more stringent growth condition on f is required.

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