LOWER BOUND FOR THE RATE OF BLOW-UP OF SINGULAR SOLUTIONS OF THE ZAKHAROV SYSTEM IN ℝ<sup>3</sup>
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Abstract
We consider the scalar Zakharov system in ℝ 3 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 [Formula: see text] with [Formula: see text]. 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.
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