-
Previous Article
Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing
- DCDS Home
- This Issue
-
Next Article
Evolution Galerkin schemes applied to two-dimensional Riemann problems for the wave equation system
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.
| [1] |
Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 |
| [2] |
Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671 |
| [3] |
Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 |
| [4] |
Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020 |
| [5] |
Pierpaolo Esposito, Maristella Petralla. Symmetries and blow-up phenomena for a Dirichlet problem with a large parameter. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1935-1957. doi: 10.3934/cpaa.2012.11.1935 |
| [6] |
Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 |
| [7] |
Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941 |
| [8] |
Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034 |
| [9] |
Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020216 |
| [10] |
Evgeny Galakhov, Olga Salieva. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Conference Publications, 2015, 2015 (special) : 489-494. doi: 10.3934/proc.2015.0489 |
| [11] |
Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 |
| [12] |
Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669 |
| [13] |
Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025 |
| [14] |
Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639 |
| [15] |
Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771 |
| [16] |
Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677 |
| [17] |
Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683 |
| [18] |
Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183 |
| [19] |
Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027 |
| [20] |
C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]