TY - JOUR

T1 - Complicated dynamics of scalar reaction diffusion equations with a nonlocal term

AU - Fiedler, Bernold

AU - Polácik, Peter

PY - 1990/1/1

Y1 - 1990/1/1

N2 - We consider the dynamics of scalar equations u, = uxx +f(x, u) + c(x)α(u), 0«x«l, where α denotes some weighted spatial average and Dirichlet boundary conditions are assumed. Prescribing f, c, α appropriately, it is shown that complicated dynamics can occur. Specifically, linearisations at equilibria can have any number of purely imaginary eigenvalues. Moreover, the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily, up to any finite order. These results are in marked contrast with the case α = 0, where bounded solutions are known to converge to equilibrium.

AB - We consider the dynamics of scalar equations u, = uxx +f(x, u) + c(x)α(u), 0«x«l, where α denotes some weighted spatial average and Dirichlet boundary conditions are assumed. Prescribing f, c, α appropriately, it is shown that complicated dynamics can occur. Specifically, linearisations at equilibria can have any number of purely imaginary eigenvalues. Moreover, the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily, up to any finite order. These results are in marked contrast with the case α = 0, where bounded solutions are known to converge to equilibrium.

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U2 - 10.1017/S0308210500024641

DO - 10.1017/S0308210500024641

M3 - Article

AN - SCOPUS:84974103874

VL - 115

SP - 167

EP - 192

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 1-2

ER -