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Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. (English) Zbl 0993.35051

The paper is concerned with stable subharmonic solutions of the time-periodic spatially inhomogeneous reaction-diffusion equation \[ u_t=\Delta u+f(z, t, u)\qquad (x\in\Omega,\;t>0) \] where \(\Omega\) is a bounded domain in a Euclidean space and \(f\) is periodic in \(t\) with period \(\tau>0\). Besides, various boundary conditions are imposed. The authors show that stable subharmonic solutions exist on any spatial domain, provided the nonlinearity is chosen suitably. By definition, a solution \(u(x, t)\) is subharmonic if it is periodic in \(t\) with the minimal period \(k\tau\) with \(k>1\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B10 Periodic solutions to PDEs
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
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