MR2976047
Nakao HAYASHI, Pavel I. NAUMKIN and Tomoyuki NIIZATO
Nakao HAYASHI, Pavel I. NAUMKIN and Tomoyuki NIIZATO
Almost Global Existence of Solutions to the Kadomtsev-Petviashvili Equations
Funkcialaj Ekvacioj. Serio Internacia
55
2012
157--168
http://fe.math.kobe-u.ac.jp/FE/FullPapers/55-1/55_157.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2976047
We consider the Cauchy problem for the Kadomtsev-Petviashvili equations $u_{t}+u_{xxx}+\sigma \partial _{x}^{-1}u_{yy}=-(u^{2})_{x}$, $(x,y)\in \mathbf{R}^{2}$, $t\in \mathbf{R}$, $u(0,x,y)=u_{0}(x,y)$, $(x,y)\in \mathbf{R}^{2}$, where $\sigma =1$ or $\sigma =-1$, $\partial _{x}^{-1}=\int_{-\infty}^{x}dx^{\prime}$. We prove that the maximal existence time $T$ is estimated from below as $T\geq \exp \left( \frac{C}{\varepsilon}\right)$, where $\varepsilon$ denotes the size of the initial data, $C>0$ is a constant.
Kadomtsev-Petviashvili equations, Almost global existence.
35Q53.
55-157
2012
Almost Global Existence of Solutions to the Kadomtsev-Petviashvili Equations
Nakao HAYASHI, Pavel I. NAUMKIN and Tomoyuki NIIZATO
Nakao HAYASHI, Pavel I. NAUMKIN and Tomoyuki NIIZATO
1
de Bouard, A.; Saut, J. C.
Solitary waves of generalized Kadomtsev-Petviashvili equations
Ann. Inst. H. Poincaré Anal. Non Linéaire
14
1997
211-236
MR1441393
2
Bourgain, J.
On the Cauchy problem for the Kadomtsev-Petviashvili equations
Geom. Funct. Anal.
3
1993
315-341
MR1223434
3
Hayashi, N.; Naumkin, P. I.; Saut, J. C.
Asymptotics for large time of global solutions to the generalized Kadomtsev-Petviashvili equation
Comm. Math. Phys.
201
1999
577-590
MR1685890
4
Iòrio, R. J.; Nunes, W. V. Leite
On equations of KP-type
Proc. Roy. Soc. Edinburgh Sect. A
128
1998
725-743
MR1635416
5
Kadmotsev, B. B.; Petviashvili, V. I.
On the stability of solitary waves in weakly dispersive media
Soviet Phys. Dokl.
15
1970
539-541
6
Niizato, T.
Large time behavior of solutions for the generalized Kadomtsev-Petviashvili equation
Differ. Equ. Appl.
3
2011
299-308
MR2830489
7
Saut, J. C.
Remarks on the generalized Kadomtsev-Petviashvili equations
Indiana Univ. Math. J.
42
1993
1011-1026
MR1254130
8
Saut, J. C.
Recent results on the generalized Kadomtsev-Petviashvili equations
Acta Appl. Math.
39
1995
477-487
MR1329578
9
Stein, E. M.
Singular Integrals and Differentiability Properties of Functions
Princeton Mathematical Series, No. 30 , Princeton Univ. Press, Princeton, NJ
1970
MR0290095
10
Tom, M. M.
On a generalized Kadomtsev-Petviashvili equation
Contemp. Math., 200, Amer. Math. Soc., Providence, RI
1996
pp.193-210
MR1410508
11
Turisyn, S. K.; Fal'kovitch, G. E.
Stability of magnetoelastic solutions and self-focusing of sound in antiferromagnet
Soviet Phys. JETP
62
1985
146-152
12
Ukai, S.
Local solutions of the Kadomtsev-Petviashvili equation
J. Fac. Sci. Univ. Tokyo Sect. IA Math.
36
1989
193-209
MR1014996
13
Zhou, X.
Inverse scattering transform for the time dependent Schrödinger equation with applications to the KP-I equation
Comm. Math. Phys.
128
1990
551-564
MR1045884