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The Riemann Problem at a Junction for a Phase Transition Traffic Model
Initial Pointwise Bounds and Blow-up for Parabolic Choquard-Pekar Inequalities
Mathematics Department, Texas A & M University, College Station, TX 77843-3368, USA |
$t \to 0^+$ |
$u\in {{C}^{2,1}}({{\mathbb{R}}^{n}}\times (0,1))\cap {{L}^{\lambda }}({{\mathbb{R}}^{n}}\times (0,1)),\ \ n\ge 1,\ \ \ \ \ \ \ \ \ \ \ \ \left( 0.1 \right)$ |
$0\le {{u}_{t}}-\Delta u\le ({{\Phi }^{\alpha /n}}*{{u}^{\lambda }}){{u}^{\sigma }}\ \ \text{ in }{{B}_{1}}\ (0)\times (0,1)\ \ \ \ \ \ \ \ \ \ \left( 0.2 \right)$ |
$α∈(0, n+2)$ |
$λ>0$ |
$σ≥0$ |
$Φ$ |
$*$ |
$\mathbb{R}^n× (0, 1)$ |
$α, λ$ |
$σ$ |
$u$ |
$B_1(0)$ |
$t \to0^+$ |
$Φ^{α/n}$ |
$Φ_α$ |
$(\frac{\partial}{\partial t}-Δ)^{α/2}$ |
References:
[1] |
H. Chen and F. Zhou, Classification of isolated singularities of positive solutions for Choquard
equations, J. Differential Equations, 261 (2016), 6668-6698, arXiv: 1512.03181.
doi: 10.1016/j.jde.2016.08.047. |
[2] |
J. T. Devreese and A. S. Alexandrov,
Advances in Polaron Physics Springer Series in Solid-State Sciences, vol. 159, Springer, 2010. |
[3] |
M. Ghergu and S. D. Taliaferro, Isolated Singularities in Partial Differential Inequalities, Cambridge University Press, 2016.
doi: 10.1017/CBO9781316481363.![]() ![]() ![]() |
[4] |
M. Ghergu and S. D. Taliaferro,
Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity, J. Differential Equations, 261 (2016), 189-217.
doi: 10.1016/j.jde.2016.03.004. |
[5] |
K. R. W. Jones,
Newtonian quantum gravity, Australian Journal of Physics, 48 (1995), 1055-1082.
doi: 10.1071/PH951055. |
[6] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.
|
[7] |
P.-L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[8] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case.Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[9] |
C. Ma, W. Chen and C. Li,
Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[10] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard
equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[11] |
M. Melgaard and F. Zongo, Multiple solutions of the quasirelativistic Choquard equation,
J. Math. Phys., 53 (2012), 033709, 12pp.
doi: 10.1063/1.3695991. |
[12] |
I. M. Moroz, R. Penrose and P. Tod,
Spherically-symmetric solutions of the SchrödingerNewton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.
doi: 10.1088/0264-9381/15/9/019. |
[13] |
V. Moroz and J. Van Schaftingen,
Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[14] |
V. Moroz and J. Van Schaftingen,
Nonexistence and optimal decay of supersolutions to
Choquard equations in exterior domains, J. Differential Equations, 254 (2013), 3089-3145.
doi: 10.1016/j.jde.2012.12.019. |
[15] |
V. Moroz and J. Van Schaftingen,
Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.
doi: 10.1007/s00526-014-0709-x. |
[16] |
S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.
![]() |
[17] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhauser, Basel, 2007.
![]() ![]() |
[18] |
S. G. Samko, Hypersingular Integrals and Their Applications, Taylor and Francis, London, 2002.
![]() ![]() |
[19] |
S. D. Taliaferro,
Initial blow-up of solutions of semilinear parabolic inequalities, J. Differential Equations, 250 (2011), 892-928.
doi: 10.1016/j.jde.2010.07.033. |
[20] |
J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations,
J. Math. Phys., 50 (2009), 012905, 22pp.
doi: 10.1063/1.3060169. |
[21] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for a
nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
show all references
References:
[1] |
H. Chen and F. Zhou, Classification of isolated singularities of positive solutions for Choquard
equations, J. Differential Equations, 261 (2016), 6668-6698, arXiv: 1512.03181.
doi: 10.1016/j.jde.2016.08.047. |
[2] |
J. T. Devreese and A. S. Alexandrov,
Advances in Polaron Physics Springer Series in Solid-State Sciences, vol. 159, Springer, 2010. |
[3] |
M. Ghergu and S. D. Taliaferro, Isolated Singularities in Partial Differential Inequalities, Cambridge University Press, 2016.
doi: 10.1017/CBO9781316481363.![]() ![]() ![]() |
[4] |
M. Ghergu and S. D. Taliaferro,
Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity, J. Differential Equations, 261 (2016), 189-217.
doi: 10.1016/j.jde.2016.03.004. |
[5] |
K. R. W. Jones,
Newtonian quantum gravity, Australian Journal of Physics, 48 (1995), 1055-1082.
doi: 10.1071/PH951055. |
[6] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.
|
[7] |
P.-L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[8] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case.Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[9] |
C. Ma, W. Chen and C. Li,
Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[10] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard
equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[11] |
M. Melgaard and F. Zongo, Multiple solutions of the quasirelativistic Choquard equation,
J. Math. Phys., 53 (2012), 033709, 12pp.
doi: 10.1063/1.3695991. |
[12] |
I. M. Moroz, R. Penrose and P. Tod,
Spherically-symmetric solutions of the SchrödingerNewton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.
doi: 10.1088/0264-9381/15/9/019. |
[13] |
V. Moroz and J. Van Schaftingen,
Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[14] |
V. Moroz and J. Van Schaftingen,
Nonexistence and optimal decay of supersolutions to
Choquard equations in exterior domains, J. Differential Equations, 254 (2013), 3089-3145.
doi: 10.1016/j.jde.2012.12.019. |
[15] |
V. Moroz and J. Van Schaftingen,
Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.
doi: 10.1007/s00526-014-0709-x. |
[16] |
S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.
![]() |
[17] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhauser, Basel, 2007.
![]() ![]() |
[18] |
S. G. Samko, Hypersingular Integrals and Their Applications, Taylor and Francis, London, 2002.
![]() ![]() |
[19] |
S. D. Taliaferro,
Initial blow-up of solutions of semilinear parabolic inequalities, J. Differential Equations, 250 (2011), 892-928.
doi: 10.1016/j.jde.2010.07.033. |
[20] |
J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations,
J. Math. Phys., 50 (2009), 012905, 22pp.
doi: 10.1063/1.3060169. |
[21] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for a
nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |


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