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2 Lu J.

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Dynamics of a modified Davey-Stewartson system in ℝ³

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Lu J.

Colloquium Mathematicum

2016

tom 145

nr 1

69-87

We study the Cauchy problem in ℝ³ for the modified Davey-Stewartson system $i∂ₜu + Δu = λ₁|u|⁴u + λ₂b₁uv_{x₁}$, $-Δv = b₂(|u|²)_{x₁}$. Under certain conditions on λ₁ and λ₂, we provide a complete picture of the local and global well-posedness, scattering and blow-up of the solutions in the energy space. Methods used in the paper are based upon the perturbation theory from [Tao et al., Comm. Partial Differential Equations 32 (2007), 1281-1343] and the convexity method from [Glassey, J. Math. Phys. 18 (1977), 1794-1797].

Blow-up for the energy-critical nonlinear wave equation and Schrödinger equation with inverse-square potential

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Lu J.

Applicationes Mathematicae

2013

tom 40

nr 2

183-196

We give a sufficient condition under which the solutions of the energy-critical nonlinear wave equation and Schrödinger equation with inverse-square potential blow up. The method is a modified variational approach, in the spirit of the work by Ibrahim et al. [Anal. PDE 4 (2011), 405-460].