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Smoothing property for the $ L^2 $-critical high-order NLS Ⅱ
Generalized linear models for population dynamics in two juxtaposed habitats
1. | Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, 76600 Le Havre, France |
2. | USTHB, LAMNEDP, Faculté de Mathématiques, BP.32, El Alia, Bab Ezzouar, 16111 Alger, Algérie |
3. | Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, 76600 Le Havre, France |
In this work we introduce a generalized linear model regulating the spread of population displayed in a $ d $-dimensional spatial region $ \Omega $ of $ \mathbb{R}^{d} $ constituted by two juxtaposed habitats having a common interface $ \Gamma $. This model is described by an operator $ \mathcal{L} $ of fourth order combining the Laplace and Biharmonic operators under some natural boundary and transmission conditions. We then invert explicitly this operator in $ L^{p} $-spaces using the $ H^{\infty } $-calculus and the Dore-Venni sums theory. This main result will lead us in a later work to study the nature of the semigroup generated by $ \mathcal{L} $ which is important for the study of the complete nonlinear generalized diffusion equation associated to it.
References:
[1] |
W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, second Edition, Monographs in Mathematics, 96, Birkhauser, 2011.
doi: 10.1007/978-3-0348-0087-7. |
[2] |
J. Bourgain,
Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 163-168.
doi: 10.1007/BF02384306. |
[3] |
D. L. Burkholder,
A geometrical characterisation of Banach spaces in which martingale dif-ference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011.
doi: 10.1214/aop/1176994270. |
[4] |
D. S. Cohen and J. D. Murray,
A generalized diffusion model for growth and dispersal in po-pulation, Journal of Mathematical Biology, 12 (1981), 237-249.
doi: 10.1007/BF00276132. |
[5] |
G. Da Prato and P. Grisvard,
Somme d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures et Appl., 54 (1975), 305-387.
|
[6] |
G. Dore and A. Venni,
On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.
doi: 10.1007/BF01163654. |
[7] |
A. Favini, R. Labbas, S. Maingot, K. Lemrabet and H. Sidibé,
Resolution and Optimal Re-gularity for a Biharmonic Equation with Impedance Boundary Conditions and Some Gene-ralizations, Discrete and Continuous Dynamical Systems, 33 (2013), 4991-5014.
doi: 10.3934/dcds.2013.33.4991. |
[8] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. |
[9] |
P. Grisvard,
Équations différentielles abstraites, Extrait des Annales scientifiques de l'École Normale Supérieure, 2 (1969), 311-395.
doi: 10.24033/asens.1178. |
[10] |
P. Grisvard,
Spazi di tracce e applicazioni, Rendiconti di Matematica, 5 (1972), 657-729.
|
[11] |
M. Haase, The Functional Calculus for Sectorial Operators, Birkhauser, 2006.
doi: 10.1007/3-7643-7698-8. |
[12] |
H. Komatsu,
Fractional powers of operators, Pacific Journal of Mathematics, 19 (1966), 285-346.
doi: 10.2140/pjm.1966.19.285. |
[13] |
R. Labbas, S. Maingot, D. Manceau and A. Thorel,
On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain, Journal of Mathematical Analysis and Applications, 450 (2017), 351-376.
doi: 10.1016/j.jmaa.2017.01.026. |
[14] |
K. Limam, R. Labbas, K. Lemrabet, A. Medeghri and M. Meisner,
On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution, Journal of Differential Equations, 259 (2015), 2695-2731.
doi: 10.1016/j.jde.2015.04.002. |
[15] |
J.-L. Lions and J. Peetre,
Sur une classe d'espaces d'interpolation, Publications Mathé-matiques de l'I.H.É.S., 19 (1964), 5-68.
|
[16] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Basel, Boston, Berlin, 1995. |
[17] |
S. G. Mikhlin,
On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR, 109 (1956), 701-703.
|
[18] |
F. L. Ochoa,
A generalized reaction-diffusion model for spatial structures formed by motile cells, BioSystems, 17 (1984), 35-50.
doi: 10.1016/0303-2647(84)90014-5. |
[19] |
J. Prüss and H. Sohr,
On operators with bounded imaginary powers in Banach spaces, Mathe-matische Zeitschrift, 203 (1990), 429-452.
doi: 10.1007/BF02570748. |
[20] |
J. Prüss and H. Sohr,
Imaginary powers of elliptic second order differential operators in Lp-spaces, Hiroshima Math. J., 23 (1993), 161-192.
doi: 10.32917/hmj/1206128381. |
[21] |
J. L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, J. Bastero, M. San Miguel (Eds.), Probability and Banach Spaces, Zaragoza, 1221 (1986), 195-222.
doi: 10.1007/BFb0099115. |
[22] |
H. Triebel, Interpolation theory, function Spaces, differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978. |
show all references
References:
[1] |
W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, second Edition, Monographs in Mathematics, 96, Birkhauser, 2011.
doi: 10.1007/978-3-0348-0087-7. |
[2] |
J. Bourgain,
Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 163-168.
doi: 10.1007/BF02384306. |
[3] |
D. L. Burkholder,
A geometrical characterisation of Banach spaces in which martingale dif-ference sequences are unconditional, Ann. Probab., 9 (1981), 997-1011.
doi: 10.1214/aop/1176994270. |
[4] |
D. S. Cohen and J. D. Murray,
A generalized diffusion model for growth and dispersal in po-pulation, Journal of Mathematical Biology, 12 (1981), 237-249.
doi: 10.1007/BF00276132. |
[5] |
G. Da Prato and P. Grisvard,
Somme d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures et Appl., 54 (1975), 305-387.
|
[6] |
G. Dore and A. Venni,
On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.
doi: 10.1007/BF01163654. |
[7] |
A. Favini, R. Labbas, S. Maingot, K. Lemrabet and H. Sidibé,
Resolution and Optimal Re-gularity for a Biharmonic Equation with Impedance Boundary Conditions and Some Gene-ralizations, Discrete and Continuous Dynamical Systems, 33 (2013), 4991-5014.
doi: 10.3934/dcds.2013.33.4991. |
[8] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. |
[9] |
P. Grisvard,
Équations différentielles abstraites, Extrait des Annales scientifiques de l'École Normale Supérieure, 2 (1969), 311-395.
doi: 10.24033/asens.1178. |
[10] |
P. Grisvard,
Spazi di tracce e applicazioni, Rendiconti di Matematica, 5 (1972), 657-729.
|
[11] |
M. Haase, The Functional Calculus for Sectorial Operators, Birkhauser, 2006.
doi: 10.1007/3-7643-7698-8. |
[12] |
H. Komatsu,
Fractional powers of operators, Pacific Journal of Mathematics, 19 (1966), 285-346.
doi: 10.2140/pjm.1966.19.285. |
[13] |
R. Labbas, S. Maingot, D. Manceau and A. Thorel,
On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain, Journal of Mathematical Analysis and Applications, 450 (2017), 351-376.
doi: 10.1016/j.jmaa.2017.01.026. |
[14] |
K. Limam, R. Labbas, K. Lemrabet, A. Medeghri and M. Meisner,
On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution, Journal of Differential Equations, 259 (2015), 2695-2731.
doi: 10.1016/j.jde.2015.04.002. |
[15] |
J.-L. Lions and J. Peetre,
Sur une classe d'espaces d'interpolation, Publications Mathé-matiques de l'I.H.É.S., 19 (1964), 5-68.
|
[16] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Basel, Boston, Berlin, 1995. |
[17] |
S. G. Mikhlin,
On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR, 109 (1956), 701-703.
|
[18] |
F. L. Ochoa,
A generalized reaction-diffusion model for spatial structures formed by motile cells, BioSystems, 17 (1984), 35-50.
doi: 10.1016/0303-2647(84)90014-5. |
[19] |
J. Prüss and H. Sohr,
On operators with bounded imaginary powers in Banach spaces, Mathe-matische Zeitschrift, 203 (1990), 429-452.
doi: 10.1007/BF02570748. |
[20] |
J. Prüss and H. Sohr,
Imaginary powers of elliptic second order differential operators in Lp-spaces, Hiroshima Math. J., 23 (1993), 161-192.
doi: 10.32917/hmj/1206128381. |
[21] |
J. L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, J. Bastero, M. San Miguel (Eds.), Probability and Banach Spaces, Zaragoza, 1221 (1986), 195-222.
doi: 10.1007/BFb0099115. |
[22] |
H. Triebel, Interpolation theory, function Spaces, differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978. |
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