Lower bound for the rate of blow-up of singular solutions of the Zakharov system in ${\mathbb{R}}^3$

Abstract

We consider the scalar Zakharov system in ${\mathbb{R}}^3$ for initial conditions $(\psi (0),n(0),n_t(0))\in H^{\ell +1/2}\times H^{\ell }\times H^{\ell -1}$, $0\le \ell \le 1$. Assuming that the solution blows up in a finite time $t^{\ast }<\infty$, we establish a lower bound for the rate of blow-up of the corresponding Sobolev norms in the form $$\Vert \psi (t){\Vert }_{H^{\ell +1/2}}+\Vert n(t){\Vert }_{H^{\ell }}+\Vert n_t(t){\Vert }_{H^{\ell -1}}>C(t^{\ast }-t{)}^{-{\theta }_{\ell }}$$ with ${\theta }_{\ell }=\frac{1}{4}(1+2\ell {)}^{-}$. The analysis is a reappraisal of the local wellposedness theory of Ginibre, Tsutsumi and Velo (1997) combined with an argument developed by Cazenave and Weissler (1990) in the context of nonlinear Schrödinger equations.

13 pages. To appear in JHDE

References in corpus (2)

• Convolutions of singular measures and applications to the Zakharov system
• Local ill-posedness of the 1D Zakharov system

Cited by in corpus (1)

• The Well-posedness and Blow-up rate of Solution for the Generalized Zakharov equations with Magnetic field in R^d