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Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity
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Characterization for the existence of bounded solutions to elliptic equations
Finite-time blowup for a Schrödinger equation with nonlinear source term
1. | Sorbonne Université & CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France |
2. | CMLS, École Polytechnique, CNRS, 91128 Palaiseau Cedex, France |
3. | Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China |
${u_t} = i\Delta u + {\left| u \right|^\alpha }u\;\;\;\;{\rm{on}}\;\;\;{\mathbb{R}^N},\;\;\alpha >0$ |
$H^1$ |
$(N-2) α ≤ 4$ |
$α≥ 2$ |
$N≤4$ |
$H^1$ |
$\mathbb{R}^N$ |
$U$ |
$U_t=|U|^α U$ |
$U(t,x)=(|x|^k-α t)^{-\frac 1α}$ |
$t<0$ |
$x∈ \mathbb{R}^N$ |
$k$ |
$U$ |
$|Δ U|\ll U_t$ |
$(t,x)=(0,0)$ |
References:
[1] |
S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Progress in Nonlinear Differential Equations and their Applications, 17. Birkhäuser Boston, Inc., Boston, MA, 1995.
doi: 10.1007/978-1-4612-2578-2. |
[2] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[3] |
T. Cazenave, S. Correia, F. Dickstein and F. B. Weissler,
A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation, São Paulo J. Math. Sci., 9 (2015), 146-161.
doi: 10.1007/s40863-015-0020-6. |
[4] |
C. Collot, T. -E. Ghoul and N. Masmoudi, Singularity formation for Burgers equation with transverse viscosity, preprint, arXiv: 1803.07826. |
[5] |
Y. Martel,
Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., 127 (2005), 1103-1140.
doi: 10.1353/ajm.2005.0033. |
[6] |
F. Merle,
Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129 (1990), 223-240.
doi: 10.1007/BF02096981. |
[7] |
F. Merle and H. Zaag,
O.D.E. type behavior of blow-up solutions of nonlinear heat equations, Current developments in partial differential equations (Temuco, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 435-450.
doi: 10.3934/dcds.2002.8.435. |
[8] |
F. Merle and H. Zaag, Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension, Séminaire: Équations aux Dérivées Partielles. 2009-2010, Exp. No. Ⅺ, 10 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2012, Available from http://sedp.cedram.org/sedp-bin/fitem?id=SEDP_2009-2010____A11_0 |
[9] |
F. Merle and H. Zaag,
On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Comm. Math. Phys., 333 (2015), 1529-1562.
doi: 10.1007/s00220-014-2132-8. |
[10] |
N. Nouaili and H. Zaag,
Construction of a blow-up solution for the complex Ginzburg-Landau equation in a critical case, Arch. Ration. Mech. Anal., 228 (2018), 995-1058.
doi: 10.1007/s00205-017-1211-3. |
[11] |
T. Ozawa,
Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 25 (2006), 403-408.
doi: 10.1007/s00526-005-0349-2. |
[12] |
P. Raphaël and J. Szeftel,
Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc., 24 (2011), 471-546.
doi: 10.1090/S0894-0347-2010-00688-1. |
[13] |
J. Speck, Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearity, preprint, arXiv: 1709.04778. |
show all references
References:
[1] |
S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Progress in Nonlinear Differential Equations and their Applications, 17. Birkhäuser Boston, Inc., Boston, MA, 1995.
doi: 10.1007/978-1-4612-2578-2. |
[2] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[3] |
T. Cazenave, S. Correia, F. Dickstein and F. B. Weissler,
A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation, São Paulo J. Math. Sci., 9 (2015), 146-161.
doi: 10.1007/s40863-015-0020-6. |
[4] |
C. Collot, T. -E. Ghoul and N. Masmoudi, Singularity formation for Burgers equation with transverse viscosity, preprint, arXiv: 1803.07826. |
[5] |
Y. Martel,
Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., 127 (2005), 1103-1140.
doi: 10.1353/ajm.2005.0033. |
[6] |
F. Merle,
Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129 (1990), 223-240.
doi: 10.1007/BF02096981. |
[7] |
F. Merle and H. Zaag,
O.D.E. type behavior of blow-up solutions of nonlinear heat equations, Current developments in partial differential equations (Temuco, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 435-450.
doi: 10.3934/dcds.2002.8.435. |
[8] |
F. Merle and H. Zaag, Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension, Séminaire: Équations aux Dérivées Partielles. 2009-2010, Exp. No. Ⅺ, 10 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2012, Available from http://sedp.cedram.org/sedp-bin/fitem?id=SEDP_2009-2010____A11_0 |
[9] |
F. Merle and H. Zaag,
On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations, Comm. Math. Phys., 333 (2015), 1529-1562.
doi: 10.1007/s00220-014-2132-8. |
[10] |
N. Nouaili and H. Zaag,
Construction of a blow-up solution for the complex Ginzburg-Landau equation in a critical case, Arch. Ration. Mech. Anal., 228 (2018), 995-1058.
doi: 10.1007/s00205-017-1211-3. |
[11] |
T. Ozawa,
Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 25 (2006), 403-408.
doi: 10.1007/s00526-005-0349-2. |
[12] |
P. Raphaël and J. Szeftel,
Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc., 24 (2011), 471-546.
doi: 10.1090/S0894-0347-2010-00688-1. |
[13] |
J. Speck, Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearity, preprint, arXiv: 1709.04778. |
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