On the Dynamics of a Degenerate Parabolic Equation: Global Bifurcation of Stationary States and Convergence

Details On the Dynamics of a Degenerate Parabolic Equation: Global Bifurcation of Stationary States and Convergence On the Dynamics of a Degenerate Parabolic Equation: Global Bifurcation of Stationary States and Convergence

2005-03-23

arxiv.org/abs/math/0503486v2

Mathematics

Analysis of PDEs

28 pages. To appear in Calculus of Variations and Partial Differential Equations. The original publication will appear at ww

Scientific paper

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.

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