Article: Infinite-time gradient blow-up in a degenerate parabolic equation Journal: International Journal of Dynamical Systems and Differential Equations (IJDSDE) 2012 Vol.4 No.1/2 pp.126 - 153 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 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). Inderscience Publishers - linking academia, business and industry through research

Title: Infinite-time gradient blow-up in a degenerate parabolic equation

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 u_t = u^pu_xx + u^q in a bounded interval Ω ⊂ ℝ, It is shown that whenever the initial data u₀ belong to W^1,∞(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 W^1,∞(Ω), for all t ∈ [0,∞), but we have ||u_x(·, t)||_L∞(Ω) → ∞ as t → ∞. Moreover, it is shown that if q 1, even the functional ∫_Ωu^α|u_x|^m blows up for some α = (m) ≥ 0. Finally, by providing explicit upper estimates for the growth of u_x 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 u_x in L^∞Ω).

Keywords: degenerate diffusion; gradient blow-up; blow-up rate; singularity formation; degenerate parabolic equations.

DOI: 10.1504/IJDSDE.2012.046001

International Journal of Dynamical Systems and Differential Equations, 2012 Vol.4 No.1/2, pp.126 - 153

Published online: 10 Dec 2014 *

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Title: Infinite-time gradient blow-up in a degenerate parabolic equation

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