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- W1963993710 abstract "Consider the equation <p align=center> $u''+|u|^{p-1}u=b|u'|^{q-1}u',quad tgeq 0,qquad $(E) <p align=left class=times> 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,<br> <em>(a)</em> if $bgeq b_1(p):=(p+1)((p+1)/2p)^{p/(p+1)}$, then all solutions blow up with a sign, with the rate <p align=center> $u(t)$~$pm (T-t)^{-2/(p-1)}quad$ as $ tto T;$ <p align=left class=times> <em>(b)</em> if $b$<$b_1(p)$, then all solutions have oscillatory blow-up, with <p align=center> $u(t)=(T-t)^{-2/(p-1)}w$(log$(T-t)+C$), <p align=left class=times> where $w(s)$ is a single sign-changing periodic function.<br> Our proofs rely on perturbed energy arguments, invariant regions and on the study of the equation for $w$ via Poincaré-Bendixson and index theory." @default.
- W1963993710 created "2016-06-24" @default.
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- W1963993710 date "2003-01-01" @default.
- W1963993710 modified "2023-09-26" @default.
- W1963993710 title "Oscillatory blow-up in nonlinear second order ODE's: The critical case" @default.
- W1963993710 doi "https://doi.org/10.3934/dcds.2003.9.577" @default.
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