BECNBIC, Número 3(Artículo 4), 2001

TÍTULO
SCALING, STABILITY AND SINGULARITIES FOR NONLINEAR, DISPERSIVE WAVE EQUATIONS: THE CRITICAL CASE (ON 28(PM)/03/2001)

AUTORES: J. Angulo¹ angulo@ime.unicamp.br, J. L. Bona², F. Linares³ and M. Scialom¹

DIRECCIÓN PARA CONTACTO
¹IMECC-UNICAMP, Caixa-Postal 6065, 13081-970, Campinas, SP, Brazil.
²Department of Mathematics and Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Austin, TX 78712 USA.
³IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil.

ENTIDADES QUE FINANCIARON LA INVESTIGACIÓN: partially suported by Grant CNPq # 300654/96-0, JA and # 300964/88-9, MA.

KEYWORDS: nonlinear dispersive waves, singularity formation, stability, similarity structure, Korteweg-de Vries-type equations, Schrödinger-type equations

For a class of generalized Korteweg-de Vries equations of the form

$\displaystyle u_t + (u^p)_x - D^\beta u_x =0 \hspace{2cm}(*)$

posed in $\mathbb{R}$ and for the focusing nonlinear Schrödinger equations

$\displaystyle iu_t + \Delta u+\vert u\vert^pu = 0 \hspace{2cm}(**)$

posed on $\mathbb{R}^n$ , it is well known that the initial-value problem is globally in time well posed provided the exponent

is less than a critical power $p_{\text{crit}}$ . For $p\geq p_{\text{crit}}$ , it is known for equation (**) and suspected for equation (*) that large initial data need not lead to globally defined solutions. It is our purpose here to investigate the critical case $p= p_{\text{crit}}$ in more detail than heretofore. Building on previous work of Weinstein, Laedke, Spatschek and their collaborators, earlier work of the present authors and others, a stability result is formulated for small perturbations of ground-state solutions of (**) and solitary-wave solutions of (*). This theorem features a scaling that is natural in the critical case. When interpreted in the contexts in view, our general result provides information about singularity formation in case the solution blows up in finite time and about large-time asymptotics in case the solution is globally defined.