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LOWER BOUND FOR THE RATE OF BLOW-UP OF SINGULAR SOLUTIONS OF THE ZAKHAROV SYSTEM IN ℝ³

Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4, Canada

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MAGDALENA CZUBAK

Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, New York 13902, USA

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, and

CATHERINE SULEM

Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4, Canada

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https://doi.org/10.1142/S0219891613500185Cited by:1 (Source: Crossref)

Abstract

We consider the scalar Zakharov system in ℝ³ for initial conditions (ψ(0), n(0), n_t(0)) ∈ H^ℓ+1/2 × H^ℓ × H^ℓ-1, 0 ≤ ℓ ≤ 1. Assuming that the solution blows up in a finite time t* < ∞, we establish a lower bound for the rate of blow-up of the corresponding Sobolev norms in the form

with

. The analysis is a reappraisal of the local well-posedness theory of Ginibre, Tsutsumi and Velo [On the Cauchy problem for the Zakharov system, J. Funct. Anal.151 (1997) 384–436] combined with an argument developed by Cazenave and Weissler [The Cauchy problem for the critical nonlinear Schrödinger equation in H^s, Nonlinear Anal.14 (1990) 807–836] in the context of nonlinear Schrödinger equations.

Keywords:

AMSC: 35Q55

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