Energy critical nls in two space dimensions
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In this article, the initial value problem for a defocusing nonlinear Schrodinger equation with exponential nonlinearity was investigated and global well-posedness was established in the subcritical and critical regimes.Abstract:
We investigate the initial value problem for a defocusing nonlinear Schrodinger equation with exponential nonlinearity $$i\partial_t u+\Delta u=u (e^{4\pi|u|^2}-1)\quad\mbox{in}\ \mathbb{R}_t\times\mathbb{R}_x^2.$$ We identify subcritical, critical, and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.read more
Citations
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The Cauchy problem for semi-linear Klein–Gordon equations in de Sitter spacetime
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Energy scattering for 2D critical wave equation
TL;DR: In this paper, the wave operators for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing exponential nonlinearity in two space dimensions are investigated, and it is shown that if the energy is below or equal to the critical value, then the solution approaches a free KleinGordon solution at the time infinity.
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Local well posedness of a 2D semilinear heat equation
TL;DR: In this paper, the authors investigated the initial value problem for a semilinear heat equation with exponential growth nonlinearity in two space dimensions, and proved the local existence and unconditional uniqueness of solutions in the Sobolev space.
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Scattering for the two-dimensional energy-critical wave equation
TL;DR: In this paper, the wave operators for the nonlinear Klein-Gordon equation with a defocusing exponential nonlinearity in two space dimensions are investigated and it is shown that if the energy is below or equal to the critical value, then the solution approaches a free Klein Gordon solution at the time infinity.
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Existence, Non-existence, and Uniqueness for a Heat Equation with Exponential Nonlinearity in ℝ2
TL;DR: In this article, the authors consider a semilinear heat equation with exponential nonlinearity in ℝ2 and prove that local solutions do not exist for certain data in the Orlicz space exp L istg 2(ℝ 2), even though a small data global existence result holds in the same space.
References
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The Cauchy problem for the critical nonlinear Schro¨dinger equation in H s
Thierry Cazenave,F. B. Weissler +1 more
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Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations
TL;DR: In this paper, the authors study defocusing analogues of these equations, namely defocusing nonlinear Schrodinger, defocusing modified Korteweg-de Vries (mKdV), and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable.
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Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R^3
TL;DR: In this article, the authors obtained global well-posedness, scattering, and global L 10 spacetime bounds for energy-class solutions to the quintic defocusing Schrodinger equa- tion in R 1+3, which is energy-critical.
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A Sharp Trudinger-Moser Type Inequality for Unbounded Domains in Rn
Yuxiang Li,Bernhard Ruf +1 more
TL;DR: In this paper, it was shown that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of ∫ Ω e 4 π u 2 dx over all such functions is uniformly bounded, independently of the domain Ω.