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On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence. (English) Zbl 1090.35035

This work is concerned with the following semilinear parabolic equation

t ϕ-div(σ(x)ϕ)-λϕ+|ϕ| 2γ ϕ=0,xΩ,t>0

under the initial-boundary conditions

ϕ(x,0)=ϕ 0 (x),xΩ,

and

ϕ| Ω =0,t>0

on a bounded or unbounded domain Ω N , N>1, which can be derived as a model for neutron diffusion. The problem is allowed to be degenerated in the sense that the diffusion coefficient σL loc 1 is allowed to be unbounded or to possess at most a finite number of essential zeros.

Under a smallness condition on γ>0 the two main results of the paper are as follows:

(1) On a σ-weighted Sobolev space as phase space, the above equation generates a semiflow with global attractor. Moreover, it is a gradient system and any solution converges to an equilibrium as t.

(2) For bounded, as well as for unbounded domains, the existence of a global branch of nonnegative stationary states for the corresponding stationary problem is established, when λ serves as bifurcation parameter.

In case of bounded domains, these results imply that any solution with nonnegative initial condition converges to the zero or the nonnegative equilibrium.

Finally, possible applications to degenerated semilinear and quasilinear elliptic equations are indicated.

MSC:
35B40Asymptotic behavior of solutions of PDE
35B41Attractors (PDE)
35R05PDEs with discontinuous coefficients or data
35K65Parabolic equations of degenerate type
35K60Nonlinear initial value problems for linear parabolic equations
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