Given any wave speed C E R, we construct a traveling wave solution of u(t) = Delta u + vertical bar del u vertical bar(2)u in an infinitely long cylinder, which connects two locally stable and axially symmetric steady states at x(3) = +/-infinity. Here u is a director field with values in S-2 subset of R-3: vertical bar u vertical bar = 1. The traveling wave has a singular point on the cylinder axis. In view of the bistable character of the potential, the result is Surprising, and it is intimately related to the nonuniqueness of the harmonic map flow itself. We show that for only one wave speed the traveling wave behaves locally, near its singular point. as a symmetric harmonic map. (C) 2009 Elsevier Inc. All rights reserved.
Bertsch, M., Primi, I. (2009). Nonuniqueness of the traveling wave speed for harmonic heat flow. JOURNAL OF DIFFERENTIAL EQUATIONS, 247(1), 69-103 [10.1016/j.jde.2009.01.003].
Nonuniqueness of the traveling wave speed for harmonic heat flow
BERTSCH, MICHIEL;
2009-01-01
Abstract
Given any wave speed C E R, we construct a traveling wave solution of u(t) = Delta u + vertical bar del u vertical bar(2)u in an infinitely long cylinder, which connects two locally stable and axially symmetric steady states at x(3) = +/-infinity. Here u is a director field with values in S-2 subset of R-3: vertical bar u vertical bar = 1. The traveling wave has a singular point on the cylinder axis. In view of the bistable character of the potential, the result is Surprising, and it is intimately related to the nonuniqueness of the harmonic map flow itself. We show that for only one wave speed the traveling wave behaves locally, near its singular point. as a symmetric harmonic map. (C) 2009 Elsevier Inc. All rights reserved.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
