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Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction
Long-time behavior of a class of nonlocal partial differential equations
1. | School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China |
2. | School of Mathematics and Statistics, Xidian University, Xi'an 710126, China |
3. | Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA |
$\begin{align*}u_t+(-Δ)^{σ/2}u +f(u) = g.\end{align*}$ |
$(L^2_0(Ω), L^2_0(Ω))$ |
$(L^2_0(Ω), H^{σ/2}_0(Ω))$ |
$\{S(t)\}_{t≥q 0}$ |
$(L^2_0(Ω), L^2_0(Ω))$ |
$(L^2_0(Ω), H^{σ/2}_0(Ω))$ |
References:
[1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. |
[2] |
I. Athanasopoulos and L. A. Caffarelli,
Continuity of the temperature in boundary heat control problems, Adv. Math., 224 (2010), 293-315.
doi: 10.1016/j.aim.2009.11.010. |
[3] |
A. Babin and M. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[4] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. |
[5] |
C. Brändle, E. Colorado, A. de Pablo and U. Sánchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Math. Roy. Soc. Edinb., 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[6] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eq., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
L. Caffarelli and A. Vasseur,
Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[8] |
Z. Chen and R. Song,
Hardy inequality for censored stable processes, Tohoku Math. J., 55 (2003), 439-450.
doi: 10.2748/tmj/1113247482. |
[9] |
R. Cont and P. Tankov, Financial Modelling With Jump Processes, Boca Raton, FL: Chapman Hall/CRC, 2004. |
[10] |
J. Duan, An Introduction to Stochastic Dynamics, Cambridge University Press, New York, 2015.
![]() ![]() |
[11] |
X. Fernández-Real and X. Ros-Oton,
Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 10 (2016), 49-64.
doi: 10.1007/s13398-015-0218-6. |
[12] |
M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Second revised and extended edition. De Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 2011. |
[13] |
P. Geredeli and A. Khanmamedov,
Long-time dynamics of the parabolic $p$-Laplacian equation, Commun. Pure Appl. Anal., 12 (2013), 735-754.
|
[14] |
A. Kiselev, F. Nazarov and A. Volberg,
Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.
doi: 10.1007/s00222-006-0020-3. |
[15] |
M. Krasnoselskii and Y. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961. |
[16] |
J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, New York: Springer-Verlag, Vol Ⅰ, 1973. |
[17] |
H. Lu, P. Bates, S. Lü and M. Zhang,
Dynamics of the 3-D fractional complex GinzburgLandau equation, J. Differ. Equ., 259 (2015), 5276-5301.
doi: 10.1016/j.jde.2015.06.028. |
[18] |
H. Lu, P. Bates, S. Lü and M. Zhang,
Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.
doi: 10.4310/CMS.2016.v14.n1.a11. |
[19] |
J. Mercado, E. Guido, A. Sánchez-Sesma, M. ͘ñiguez and A. González, Analysis of the Blasius Formula and the Navier-Stokes Fractional Equation, Chapter Fluid Dynamics in Physics, Engineering and Environmental Applications Part of the series Environmental Science and Engineering, (2012), 475–480. |
[20] |
R. Metzler and J. Klafter,
The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Mathematical and General, 37 (2004), 161-208.
doi: 10.1088/0305-4470/37/31/R01. |
[21] |
E. Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. sci. math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[22] |
A. de Pablo, F. Quirós, A. Rodriguez and J. Vázquez,
A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.
doi: 10.1016/j.aim.2010.07.017. |
[23] |
A. de Pablo, F. Quirós, A. Rodriguez and J. Vázquez,
A general fractional porous medium
equation, Comm. Pure Applied Math., 65 (2012), 1242-1284.
doi: 10.1002/cpa.21408. |
[24] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[25] |
J. Simon,
Compact sets in the space Lp(O, T; B), Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.
|
[26] |
R. Song and Z. Vondraček,
Potential theory of subordinate killed Brownian motion in a
domain, Probab. Theory Relat. Fields, 125 (2003), 578-592.
doi: 10.1007/s00440-002-0251-1. |
[27] |
P. Stinga and J. Torrea,
Extension problem and Harnack's inequality for some fractional operators, Commun. Partial Differ. Equ., 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[28] |
J. Vázquez,
Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.
doi: 10.3934/dcdss.2014.7.857. |
[29] |
M. Yang, C. Sun and C. Zhong,
Global attractors for p-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142.
doi: 10.1016/j.jmaa.2006.04.085. |
[30] |
X. Zhang,
Stochastic lagrangian particle approach to fractal Navier-Stokes equations, Commun. Math. Phys., 311 (2012), 133-155.
doi: 10.1007/s00220-012-1414-2. |
[31] |
C. Zhang, J. Zhang and C. Zhong,
Existence of weak solutions for fractional porous medium equations with nonlinear term, Appl. Math. Lett., 61 (2016), 95-101.
doi: 10.1016/j.aml.2016.05.001. |
[32] |
C. Zhong, M. Yang and C. Sun,
The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differ. Equ., 223 (2006), 367-399.
doi: 10.1016/j.jde.2005.06.008. |
show all references
References:
[1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. |
[2] |
I. Athanasopoulos and L. A. Caffarelli,
Continuity of the temperature in boundary heat control problems, Adv. Math., 224 (2010), 293-315.
doi: 10.1016/j.aim.2009.11.010. |
[3] |
A. Babin and M. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[4] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. |
[5] |
C. Brändle, E. Colorado, A. de Pablo and U. Sánchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Math. Roy. Soc. Edinb., 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[6] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eq., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
L. Caffarelli and A. Vasseur,
Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[8] |
Z. Chen and R. Song,
Hardy inequality for censored stable processes, Tohoku Math. J., 55 (2003), 439-450.
doi: 10.2748/tmj/1113247482. |
[9] |
R. Cont and P. Tankov, Financial Modelling With Jump Processes, Boca Raton, FL: Chapman Hall/CRC, 2004. |
[10] |
J. Duan, An Introduction to Stochastic Dynamics, Cambridge University Press, New York, 2015.
![]() ![]() |
[11] |
X. Fernández-Real and X. Ros-Oton,
Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 10 (2016), 49-64.
doi: 10.1007/s13398-015-0218-6. |
[12] |
M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Second revised and extended edition. De Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 2011. |
[13] |
P. Geredeli and A. Khanmamedov,
Long-time dynamics of the parabolic $p$-Laplacian equation, Commun. Pure Appl. Anal., 12 (2013), 735-754.
|
[14] |
A. Kiselev, F. Nazarov and A. Volberg,
Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.
doi: 10.1007/s00222-006-0020-3. |
[15] |
M. Krasnoselskii and Y. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961. |
[16] |
J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, New York: Springer-Verlag, Vol Ⅰ, 1973. |
[17] |
H. Lu, P. Bates, S. Lü and M. Zhang,
Dynamics of the 3-D fractional complex GinzburgLandau equation, J. Differ. Equ., 259 (2015), 5276-5301.
doi: 10.1016/j.jde.2015.06.028. |
[18] |
H. Lu, P. Bates, S. Lü and M. Zhang,
Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.
doi: 10.4310/CMS.2016.v14.n1.a11. |
[19] |
J. Mercado, E. Guido, A. Sánchez-Sesma, M. ͘ñiguez and A. González, Analysis of the Blasius Formula and the Navier-Stokes Fractional Equation, Chapter Fluid Dynamics in Physics, Engineering and Environmental Applications Part of the series Environmental Science and Engineering, (2012), 475–480. |
[20] |
R. Metzler and J. Klafter,
The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Mathematical and General, 37 (2004), 161-208.
doi: 10.1088/0305-4470/37/31/R01. |
[21] |
E. Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. sci. math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[22] |
A. de Pablo, F. Quirós, A. Rodriguez and J. Vázquez,
A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.
doi: 10.1016/j.aim.2010.07.017. |
[23] |
A. de Pablo, F. Quirós, A. Rodriguez and J. Vázquez,
A general fractional porous medium
equation, Comm. Pure Applied Math., 65 (2012), 1242-1284.
doi: 10.1002/cpa.21408. |
[24] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[25] |
J. Simon,
Compact sets in the space Lp(O, T; B), Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.
|
[26] |
R. Song and Z. Vondraček,
Potential theory of subordinate killed Brownian motion in a
domain, Probab. Theory Relat. Fields, 125 (2003), 578-592.
doi: 10.1007/s00440-002-0251-1. |
[27] |
P. Stinga and J. Torrea,
Extension problem and Harnack's inequality for some fractional operators, Commun. Partial Differ. Equ., 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[28] |
J. Vázquez,
Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.
doi: 10.3934/dcdss.2014.7.857. |
[29] |
M. Yang, C. Sun and C. Zhong,
Global attractors for p-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142.
doi: 10.1016/j.jmaa.2006.04.085. |
[30] |
X. Zhang,
Stochastic lagrangian particle approach to fractal Navier-Stokes equations, Commun. Math. Phys., 311 (2012), 133-155.
doi: 10.1007/s00220-012-1414-2. |
[31] |
C. Zhang, J. Zhang and C. Zhong,
Existence of weak solutions for fractional porous medium equations with nonlinear term, Appl. Math. Lett., 61 (2016), 95-101.
doi: 10.1016/j.aml.2016.05.001. |
[32] |
C. Zhong, M. Yang and C. Sun,
The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differ. Equ., 223 (2006), 367-399.
doi: 10.1016/j.jde.2005.06.008. |
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