Approaching Critical Decay in a Strongly Degenerate Parabolic Equation
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Approaching Critical Decay in a Strongly Degenerate Parabolic Equation Michael Winkler1
Received: 27 June 2020 / Revised: 15 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The Cauchy problem in Rn , n ≥ 1, for the parabolic equation u t = u p u
()
is considered in the strongly degenerate regime p ≥ 1. The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies 1
t p u(·, t) L ∞ (Rn ) → ∞ as t → ∞.
(0.1)
The first result of this study complements this by asserting that given any positive f ∈ C 0 ([0, ∞)) fulfilling f (t) → +∞ as t → ∞ one can find a positive nondecreasing function φ ∈ C 0 ([0, ∞)) such that whenever u 0 ∈ C 0 (Rn ) is radially symmetric with 0 < u 0 < φ(| · |), the corresponding minimal solution u satisfies 1
t p u(·, t) L ∞ (Rn ) → 0 as t → ∞. f (t) Secondly, () is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data u 0 . It is shown that if the connected components of {u 0 > 0} comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to () satisfies 1 1 0 < lim inf t p u(·, t) L ∞ (Rn ) ≤ lim sup t p u(·, t) L ∞ (Rn ) < ∞. t→∞
t→∞
Under a somewhat complementary hypothesis, particularly fulfilled if {u 0 > 0} contains components with arbitrarily small principal eigenvalues of the associated Dirichlet Laplacian, it is finally seen that (0.1) continues to hold also for such not everywhere positive weak solutions.
B 1
Michael Winkler [email protected] Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany
123
Journal of Dynamics and Differential Equations
Keywords Degenerate parabolic equation · Decay rates of solutions Mathematics Subject Classification 35B40 (Primary) · 35K65 (Secondary)
1 Introduction The dynamical features of the nonlinear parabolic equation u t = u p u
(1.1)
are known to depend quite crucially on the exponent p > 0 that quantifies the strength of diffusion degeneracies in regions where the solution is small; indeed, a considerable literature has rigorously revealed various parabolictiy-diminishing effects going along with an increase of p. Among the most comprehensively understood aspects in this regard seem to be phenomena related to propagation of positivity: In striking difference to the borderline case p = 0 of the linear heat equation, throughout the range p ∈ (0, 1) in which (1.1) is equivalent to the porous medium equation vt = v m with m = 1−1 p > 1, compactly supported initial data evolve into continuous solutions [8] which at each point in the considered domain do eventually become positive, but the spatial positivity set of which propagates at finite speed ([5,15]; see also [2,7,9,13] for more detailed information, and [1] or [16] for an overview). In this respect
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