Authors: Michael Winkler
Addresses: Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany
Abstract: This work deals with the Dirichlet problem for the degenerate parabolic equation ut = upuxx + uq in a bounded interval Ω ⊂ ℝ, It is shown that whenever the initial data u0 belong to W1,∞(Ω), are nonnegative and vanish on ∂Ω, the so-called maximal solution u undergoes an infinite-time gradient blow-up. That is, the function u(·, t) belongs to W1,∞(Ω), for all t ∈ [0,∞), but we have ||ux(·, t)||L∞(Ω) → ∞ as t → ∞. Moreover, it is shown that if q < p 1 then for sufficiently large m > 1, even the functional ∫Ωuα|ux|m blows up for some α = (m) ≥ 0. Finally, by providing explicit upper estimates for the growth of ux with respect to time, it is shown that the rate of gradient blow-up in any of the integral norms considered above is not faster than algebraic, provided that q > 1. In the special case when u0(x) ≥ cdist(x, ∂Ω) for all x ∈ Ω and some c > 0, the same is valid for the norm of ux in L∞Ω).
Keywords: degenerate diffusion; gradient blow-up; blow-up rate; singularity formation; degenerate parabolic equations.
International Journal of Dynamical Systems and Differential Equations, 2012 Vol.4 No.1/2, pp.126 - 153
Available online: 19 Mar 2012 *Full-text access for editors Access for subscribers Purchase this article Comment on this article