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Register a new userL^2-concentration phenomenon for Zakharov system below energy norm
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In this paper, well prove a L^2-concentration result of Zakharov system in space dimension two, with radial initial data (u_0,n_0,n_1)in H^stimes L^2times H^{-1} ({16/17}<s<1), when blow up of the solution happens by I-method. In additional to that we find a blow up character of this system. Furthermore, we improve the global existence result of Bourgains to above spaces.
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We prove dynamical dichotomy into scattering and blow-up (in a weak sense) for all radial solutions of the Zakharov system in the energy space of four spatial dimensions that have less energy than the ground state, which is written using the Aubin-Talenti function. The dichotomy is characterized by the critical mass of the wave component of the ground state. The result is similar to that by Kenig and Merle for the energy-critical nonlinear Schrodinger equation (NLS). Unlike NLS, however, the most difficult interaction in the proof stems from the free wave component. In order to control it, the main novel ingredient we develop in this paper is a uniform global Strichartz estimate for the linear Schrodinger equation with a potential of subcritical mass solving a wave equation. This estimate, as well as the proof, may be of independent interest. For the scattering proof, we follow the idea by Dodson and Murphy.
We consider the global dynamics below the ground state energy for the Zakharov system in the 3D radial case. We obtain dichotomy between the scattering and the growup.
We prove scattering for the radial nonlinear Klein-Gordon equation $ partial_{tt} u - Delta u + u = -|u|^{p-1} u $ with $5 > p >3$ and data $ (u_{0}, u_{1}) in H^{s} times H^{s-1} $, $ 1 > s > 1- frac{(5-p)(p-3)}{2(p-1)(p-2)} $ if $ 4 geq p > 3 $ and $ 1 > s > 1 - frac{(5-p)^{2}}{2(p-1)(6-p)}$ if $ 5> p geq 4$. First we prove Strichartz-type estimates in $ L_{t}^{q} L_{x}^{r} $ spaces. Then by using these decays we establish some local bounds. By combining these results with a Morawetz-type estimate and a radial Sobolev inequality we control the variation of an almost conserved quantity on arbitrarily large intervals. Once we have showed that this quantity is controlled, we prove that some of these local bounds can be upgraded to global bounds. This is enough to establish scattering. All the estimates involved require a delicate analysis due to the nature of the nonlinearity and the lack of scaling.
We prove small energy scattering for the 3D Zakharov system with radial symmetry. The main ingredients are normal form reduction and the radial-improved Strichartz estimates.
The two-dimensional Zakharov system is shown to have a unique global solution for data without finite energy if the L^2 - norm of the Schrodinger part is small enough. The proof uses a refined I-method originally initiated by Colliander, Keel, Staffilani, Takaoka and Tao. A polynomial growth bound for the solution is also given.
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