On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence |
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Authors: | Nikos I Karachalios Nikos B Zographopoulos |
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Institution: | (1) Department of Mathematics, University of the Aegean, Karlovassi, GR 83200 Samos, Greece;(2) Department of Applied Mathematics, University of Crete, GR 71409 Athens, Greece |
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Abstract: | We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which
may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering
both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with
the definition of a gradient dynamical system in the natural phase space-that at least in the case of a bounded domain, any
solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium. Applications of the global bifurcation
result to general degenerate semilinear as well as to quasilinear elliptic equations, are also discussed.
Mathematics Subject Classification (1991) 35B40, 35B41, 35R05 |
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Keywords: | Degenerate parabolic equation Global attractor Global bifurcation Generalized sobolev spaces |
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