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Oscillatory blow-up in nonlinear second order ODE's: The critical case
| 1. | Laboratoire Analyse, Géométrie et Applications, UMR CNRS 7539, Institut Galillée, Université Paris Nord, 93430 Villetaneuse, France |
| 2. | Departement of Mathematics, Lebanese University, P.O.box 155/012 Beirut, Lebanon |
| 3. | Département de Mathématiques, Université de Picardie, INSSET, 02109 St-Quentin |
$u''+|u|^{p-1}u=b|u'|^{q-1}u',\quad t\geq 0,\qquad $(E)
where $p$, $q>1$ and $b>0$ are real numbers. A detailed study of
the large-time behavior
of solutions of (E) was carried out in [5]. We here investigate
the critical case $q=2p/(p+1)$, which is
scale-invariant and was not covered in [5]. We prove that all
nontrivial solutions blow-up in finite time and
that the asymptotic behavior near blow-up exhibits a strong
dependence upon the values of $b$. Namely,
(a) if $b\geq b_1(p):=(p+1)((p+1)/2p)^{p/(p+1)}$,
then all solutions blow
up with a sign, with the rate
$u(t)$~$\pm (T-t)^{-2/(p-1)}\quad$ as $ t\to T;$
(b) if $b$<$b_1(p)$, then all solutions have oscillatory blow-up, with
$u(t)=(T-t)^{-2/(p-1)}w$(log$(T-t)+C$),
where $w(s)$
is a single sign-changing periodic function.
Our proofs
rely on perturbed energy arguments, invariant regions
and on the study of the equation for $w$ via Poincaré-Bendixson
and index theory.
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