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October  2017, 37(10): 5211-5252. doi: 10.3934/dcds.2017226

Initial Pointwise Bounds and Blow-up for Parabolic Choquard-Pekar Inequalities

Steven D. Taliaferro

Mathematics Department, Texas A & M University, College Station, TX 77843-3368, USA

Received  December 2016 Revised  May 2017 Published  June 2017

We study the behavior as
$t \to 0^+$
 of nonnegative functions
$u\in {{C}^{2,1}}({{\mathbb{R}}^{n}}\times (0,1))\cap {{L}^{\lambda }}({{\mathbb{R}}^{n}}\times (0,1)),\ \ n\ge 1,\ \ \ \ \ \ \ \ \ \ \ \ \left( 0.1 \right)$
satisfying the parabolic Choquard-Pekar type inequalities
$0\le {{u}_{t}}-\Delta u\le ({{\Phi }^{\alpha /n}}*{{u}^{\lambda }}){{u}^{\sigma }}\ \ \text{ in }{{B}_{1}}\ (0)\times (0,1)\ \ \ \ \ \ \ \ \ \ \left( 0.2 \right)$
where
$α∈(0, n+2)$
,
$λ>0$
, and
$σ≥0$
 are constants,
$Φ$
 is the heat kernel, and
$*$
 is the convolution operation in
$\mathbb{R}^n× (0, 1)$
. We provide optimal conditions on
$α, λ$
, and
$σ$
 such that nonnegative solutions
$u$
 of (0.1), (0.2) satisfy pointwise bounds in compact subsets of
$B_1(0)$
 as
$t \to0^+$
. We obtain similar results for nonnegative solutions of (0.1), (0.2) when
$Φ^{α/n}$
 in (0.2) is replaced with the fundamental solution
$Φ_α$
 of the fractional heat operator
$(\frac{\partial}{\partial t}-Δ)^{α/2}$
.
Keywords: Nonlocal, parabolic, pointwise bound, initial blow-up, Choquard, heat potential.
Mathematics Subject Classification: 35B09, 35B33, 35B44, 35B45, 35K10, 35K58, 35R09, 35R45.
Citation: Steven D. Taliaferro. Initial Pointwise Bounds and Blow-up for Parabolic Choquard-Pekar Inequalities. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5211-5252. doi: 10.3934/dcds.2017226
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.

Figure 1.  Case $\alpha\in (2,n+2)$
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Figure 2.  Case $\alpha\in (0,2]$. When $\alpha=2$ the graph on the interval $\lambda>(n+2)/n$ is the horizontal half line $\sigma=1$
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