CONSTRUCTION OF MULTIBUBBLE SOLUTIONS FOR THE CRITICAL GKDV EQUATION

被引:11
作者
Combet, Vianney [1 ]
Martel, Yvan [2 ]
机构
[1] Univ Lille, CNRS, UMR 8524, Lab Paul Painleve, F-59000 Lille, France
[2] Univ Paris Saclay, CNRS, Ecole Polytech, CMLS, F-91128 Palaiseau, France
关键词
gKdV; blow up; multibubbles; NONLINEAR SCHRODINGER-EQUATION; BLOW-UP SOLUTIONS; GENERALIZED KDV EQUATION; SEMILINEAR WAVE-EQUATION; ONE SPACE DIMENSION; DE-VRIES-EQUATIONS; MINIMAL MASS; MULTISOLITON SOLUTIONS; SUPERCRITICAL GKDV; NLS EQUATIONS;
D O I
10.1137/17M1140595
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation partial derivative(t)u+partial derivative(x)(partial derivative(xx)u+u(5)) = 0 containing an arbitrary number K >= 2 of blow up bubbles for any choice of sign and scaling parameters: for any l(1) > l(2) > ... > l(K) > 0 and c(1),..., c(K) is an element of {+/- 1}, there exists an H-1 solution u of the equation such that u(t) - Sigma(K)(k=1) epsilon(k)/lambda(1/2)(k) Q(-x(k)(t)/lambda(k)(t)) -> 0 in H-1 as t down arrow 0 with lambda(k)(t) similar to l(k)t and x(k)(t) similar to -l(k)(-2)t(-1) as t down arrow 0. The construction uses and extends techniques developed mainly in [ Y. Martel and F. Merle, J. Math. Pures Appl. (9), 79 (2000), pp. 339-425] and [Y. Martel, F. Merle, and P. Raphael, Acta Math., 212 (2014), pp. 59-140; J. Eur. Math. Soc. (JEMS), 17 (2015), pp. 1855-1925; Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14 (2015), pp. 575-631]. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved in [V. Combet and Y. Martel, Bull. Sci. Math., 141 (2017), pp. 20-103].
引用
收藏
页码:3715 / 3790
页数:76
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