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Oscillatory blow-up in nonlinear second order ODE's: The critical case

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  • Consider the equation

    $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.

    Mathematics Subject Classification: 34C11, 34C15, 34A12.

    Citation:

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