On the Dynamics of a Degenerate Parabolic Equation: Global Bifurcation of Stationary States and Convergence
Nikos I. Karachalios, Nikos B. Zographopoulos
TL;DR
The paper investigates a degenerate parabolic equation with a diffusion σ(x) that may vanish or blow up, on bounded and unbounded domains. It develops a gradient dynamical-systems framework in the weighted energy space $D^{1,2}_0(\Omega; \sigma)$ with a Lyapunov functional $\mathcal{J}$, establishing a global attractor and a global branch of nonnegative stationary states bifurcating from the principal eigenvalue $\lambda_1$, using nondegenerate approximations ($(P)_r$, $(P)_R$). In bounded domains, nonnegative initial data converge to the trivial equilibrium for $\lambda<\lambda_1$ or to the unique nonnegative equilibrium on the branch $C_{\lambda_1}$ for $\lambda>\lambda_1$ (a supercritical pitchfork); in unbounded domains, a weighted-energy framework yields asymptotic compactness and a global attractor via domain-approximation arguments. The global bifurcation results extend to general degenerate semilinear and quasilinear elliptic equations under natural hypotheses, with the principal eigenvalue as a bifurcation point and a global branch of nonnegative solutions, and with a degeneracy condition $0<\alpha\le\alpha^*$ (\alpha^* = 2(1-\gamma(N-2))/(2\gamma+1)) ensuring diffusion control and convergence.
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.