The initial value problem for the quadratic nonlinear Klein-Gordon equation.
Hayashi, Nakao, Naumkin, Pavel I. (2010)
Advances in Mathematical Physics
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Hayashi, Nakao, Naumkin, Pavel I. (2010)
Advances in Mathematical Physics
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Kaikina, Elena I. (2006)
International Journal of Mathematics and Mathematical Sciences
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Sahbi Keraani, Ana Vargas (2009)
Annales de l'I.H.P. Analyse non linéaire
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Galaktionov, Victor A. (2003)
International Journal of Mathematics and Mathematical Sciences
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Frank Merle, Pierre Raphaël, Jérémie Szeftel (2009-2010)
Séminaire Équations aux dérivées partielles
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We consider the focusing nonlinear Schrödinger equations . We prove the existence of two finite time blow up dynamics in the supercritical case and provide for each a qualitative description of the singularity formation near the blow up time.
V. Banica, L. Vega (2008)
Annales de l'I.H.P. Analyse non linéaire
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Messaoudi, Salim A., Said-Houari, Belkacem (2004)
Journal of Applied Mathematics
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Axel Grünrock (2010)
Open Mathematics
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The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces defined by the norm . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ . The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0-uniform sense, if s < . The results for r =...