On Nonexistence of Baras – Goldstein Type for Higher-order Parabolic Equations with Singular Potentials

Content Provider	Semantic Scholar
Author	Kamotski, Ilia V.
Copyright Year	2009
Abstract	The celebrated result by Baras and Goldstein (1984) established that the heat equation with singular inverse square potential in a smooth bounded domain Ω ⊂ RN , N ≥ 3, such that 0 ∈ Ω, ut = Δu+ c |x|2 u in Ω× (0, T ), u ∣∣ ∂Ω = 0, in the supercritical range c > cHardy(1) = ( N−2 2 )2 , does not have a solution for any nontrivial L1 initial data u0(x) ≥ 0 in Ω or for a positive measure. Namely, it was proved that a regular approximation of a possible solution by a sequence {un(x, t)} of classical solutions of uniformly parabolic equations with bounded truncated potentials given by V (x) = c |x|2 → Vn(x) = min { c |x|2 , n } (n ≥ 1) diverges, and, as n → ∞, un(x, t) → +∞ in Ω× (0, T ). In the present paper, we reveal the connection of this “very singular” evolution with a spectrum of some “limiting” operator. The proposed approach allows us to consider more general higher-order operators (for which Hardy’s inequalities were known since Rellich, 1954) and initial data that are not necessarily positive. In particular it is established that, under some natural hypothesis, the divergence result is valid for any 2mth-order parabolic equation with singular potential ut = −(−Δ)mu+ c |x|2m u in Ω× (0, T ), where c > cH(m), m ≥ 1, with zero Dirichlet conditions on ∂Ω and for a wide class of initial data. In particular, typically, the divergence holds for any data satisfying u0(x) is continuous at x = 0 and u0(0) > 0. Similar nonexistence (i.e., divergence as ε → 0) results are also derived for time-dependent potentials ε−2mq(x ε , t ε2m ) and nonlinear reaction terms |u| ε2m+|x|2m with p>1. Applications to other, linear and semilinear, Schrödinger and wave PDEs are discussed. Received by the editors February 4, 2008. 2000 Mathematics Subject Classification. Primary 35K55, 35K40.
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Alternate Webpage(s)	http://www.ams.org/journals/tran/2010-362-08/S0002-9947-10-04855-5/S0002-9947-10-04855-5.pdf
Alternate Webpage(s)	https://purehost.bath.ac.uk/ws/portalfiles/portal/288064/TransAMS_362_8_4117.pdf
Alternate Webpage(s)	http://arxiv.org/pdf/0901.4171v1.pdf
Alternate Webpage(s)	http://www.ams.org/tran/0000-000-00/S0002-9947-10-04855-5/S0002-9947-10-04855-5.pdf
Language	English
Access Restriction	Open
Subject Keyword	Approximation Arabic numeral 0 Commutation theorem Emoticon Mathematics Subject Classification Maxima and minima Nonlinear system Parabolic antenna Rectangular potential barrier Schrödinger Singular Solutions Vergence
Content Type	Text
Resource Type	Article