doi: 10.3934/dcdss.2021028
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Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces

1. 

Université Clermont Auvergne CNRS, LMBP F-63000 Clermont-Ferrand, France

2. 

Université Sultan Moulay Slimane, Faculté des Sciences et Techniques, Laboratoire de Mathématiques et Applications, B.P. 523, 23000 Beni-Mellal, Morocco

3. 

Université Abdelmalek Essâadi, Faculté des Sciences et Techniques, Laboratoire de Mathématiques et Applications, B.P. 416, 90000 Tangier, Morocco

* Corresponding author: Ahmed Zeghal (azeghal@uae.ac.ma)

Received  August 2020 Revised  January 2021 Early access March 2021

Fund Project: The second author is supported by CNRST grant 18USMS2016

We establish some results regarding the existence of solutions to a nonlinear mono-energetic singular transport equation in slab geometry on $ L^p $-spaces with $ p\in (1,+\infty) $. Both the cases where the boundary conditions are specular reflections and periodic are discussed.

Citation: Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021028
References:
[1]

J. Appell, The superposition operator in function spaces – A survey, Exposition Math., 6 (1988), 209-270.   Google Scholar

[2] J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511897450.  Google Scholar
[3]

R. Beals and V. Protopopescu, Abstract time-dependent transport equations, J. Math. Anal. Appl., 121 (1987), 370-405.  doi: 10.1016/0022-247X(87)90252-6.  Google Scholar

[4]

M. Cessenat, Théorèmes de trace $L^p$ pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 831-834.   Google Scholar

[5]

M. Cessenat, Théorèmes de trace pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér I Math., 300 (1985), 89-92.   Google Scholar

[6]

M. Chabi, Théorie de scattering dans les espaces de Banach réticulés. Transport singulier dans $L^1$, Thèse de doctorat, Université de Franche-Comté, 1995. Google Scholar

[7]

M. Chabi and K. Latrach, On singular mono-energetic transport equations in slab geometry, Math. Methods Appl. Sci., 25 (2002), 1121-1147.  doi: 10.1002/mma.330.  Google Scholar

[8]

M. Chabi and K. Latrach, Singular one-dimensional transport equations on $L_p$-spaces, J. Math. Anal. Appl., 283 (2003), 319-336.  doi: 10.1016/S0022-247X(03)00299-3.  Google Scholar

[9]

P. Dràbek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter & Co., Berlin, 1997. doi: 10.1515/9783110804775.  Google Scholar

[10]

J. Garcia-FalsetK. Latrach and A. Zeghal, Existence and uniqueness results for a nonlinear evolution equation arising in growing cell populations, Nonlinear Anal., 97 (2014), 210-227.  doi: 10.1016/j.na.2013.11.027.  Google Scholar

[11]

K. J{ö}rgens, Linear Integral Operators, Pitman, Advanced Publishing Program, Boston, 1982.  Google Scholar

[12]

M. A. Krasnoselskii, et al., Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden, 1976.  Google Scholar

[13]

K. Latrach, Introduction à la Théorie de Points Fixes Métrique et Topologique avec Apllications, Edition Ellipses, Collection Références Sciences, 2017. Google Scholar

[14]

K. Latrach, H. Oummi and A. Zeghal, Existence results for nonlinear mono-energetic singular transport equations: $L^1$-spaces, Mediterr. J. Math., 16 (2019), 22 pp. doi: 10.1007/s00009-018-1282-x.  Google Scholar

[15]

K. LatrachM. A. Taoudi and A. Zeghal, Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations, J. Differential Equations, 221 (2006), 256-271.  doi: 10.1016/j.jde.2005.04.010.  Google Scholar

[16]

K. Latrach and A. Zeghal, Existence results for a nonlinear boundary value problem arising in growing cell populations, Math. Models Methods Appl. Sci., 13 (2003), 1-17.  doi: 10.1142/S0218202503002350.  Google Scholar

[17]

B. Lods, On linear kinetic equations involving unbounded cross-sections, Math. Methods Appl. Sci., 27 (2004), 1049-1075.  doi: 10.1002/mma.485.  Google Scholar

[18]

M. Mokhtar-Kharroubi, Time asymptotic behaviour and compactness in neutron transport theory, European J. of Mech. B Fluids, 11 (1992), 39-68.   Google Scholar

[19]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. New Aspects, Series on advances in Mathematics for Applied Sciences, 46, World Scientific, 1997. doi: 10.1142/3288.  Google Scholar

[20]

M. Mokhtar-Kharroubi, Optimal spectral theory of the linear Boltzmann equation, J. Funct. Anal., 226 (2005), 21-47.  doi: 10.1016/j.jfa.2005.02.014.  Google Scholar

[21]

B. Montagnini and M. L. Demuru, Complete continuity of the free gas scattering operator in neutron thermalization theory, J. Math. Anal. Appl., 12 (1965), 49-57.  doi: 10.1016/0022-247X(65)90052-1.  Google Scholar

[22] D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1974.   Google Scholar
[23]

A. Suhadolc, Linearized Boltzmann equation in $L^1$ space, J. Math. Anal. Appl., 35 (1971), 1-13.  doi: 10.1016/0022-247X(71)90231-9.  Google Scholar

show all references

References:
[1]

J. Appell, The superposition operator in function spaces – A survey, Exposition Math., 6 (1988), 209-270.   Google Scholar

[2] J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511897450.  Google Scholar
[3]

R. Beals and V. Protopopescu, Abstract time-dependent transport equations, J. Math. Anal. Appl., 121 (1987), 370-405.  doi: 10.1016/0022-247X(87)90252-6.  Google Scholar

[4]

M. Cessenat, Théorèmes de trace $L^p$ pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 831-834.   Google Scholar

[5]

M. Cessenat, Théorèmes de trace pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér I Math., 300 (1985), 89-92.   Google Scholar

[6]

M. Chabi, Théorie de scattering dans les espaces de Banach réticulés. Transport singulier dans $L^1$, Thèse de doctorat, Université de Franche-Comté, 1995. Google Scholar

[7]

M. Chabi and K. Latrach, On singular mono-energetic transport equations in slab geometry, Math. Methods Appl. Sci., 25 (2002), 1121-1147.  doi: 10.1002/mma.330.  Google Scholar

[8]

M. Chabi and K. Latrach, Singular one-dimensional transport equations on $L_p$-spaces, J. Math. Anal. Appl., 283 (2003), 319-336.  doi: 10.1016/S0022-247X(03)00299-3.  Google Scholar

[9]

P. Dràbek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter & Co., Berlin, 1997. doi: 10.1515/9783110804775.  Google Scholar

[10]

J. Garcia-FalsetK. Latrach and A. Zeghal, Existence and uniqueness results for a nonlinear evolution equation arising in growing cell populations, Nonlinear Anal., 97 (2014), 210-227.  doi: 10.1016/j.na.2013.11.027.  Google Scholar

[11]

K. J{ö}rgens, Linear Integral Operators, Pitman, Advanced Publishing Program, Boston, 1982.  Google Scholar

[12]

M. A. Krasnoselskii, et al., Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden, 1976.  Google Scholar

[13]

K. Latrach, Introduction à la Théorie de Points Fixes Métrique et Topologique avec Apllications, Edition Ellipses, Collection Références Sciences, 2017. Google Scholar

[14]

K. Latrach, H. Oummi and A. Zeghal, Existence results for nonlinear mono-energetic singular transport equations: $L^1$-spaces, Mediterr. J. Math., 16 (2019), 22 pp. doi: 10.1007/s00009-018-1282-x.  Google Scholar

[15]

K. LatrachM. A. Taoudi and A. Zeghal, Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations, J. Differential Equations, 221 (2006), 256-271.  doi: 10.1016/j.jde.2005.04.010.  Google Scholar

[16]

K. Latrach and A. Zeghal, Existence results for a nonlinear boundary value problem arising in growing cell populations, Math. Models Methods Appl. Sci., 13 (2003), 1-17.  doi: 10.1142/S0218202503002350.  Google Scholar

[17]

B. Lods, On linear kinetic equations involving unbounded cross-sections, Math. Methods Appl. Sci., 27 (2004), 1049-1075.  doi: 10.1002/mma.485.  Google Scholar

[18]

M. Mokhtar-Kharroubi, Time asymptotic behaviour and compactness in neutron transport theory, European J. of Mech. B Fluids, 11 (1992), 39-68.   Google Scholar

[19]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. New Aspects, Series on advances in Mathematics for Applied Sciences, 46, World Scientific, 1997. doi: 10.1142/3288.  Google Scholar

[20]

M. Mokhtar-Kharroubi, Optimal spectral theory of the linear Boltzmann equation, J. Funct. Anal., 226 (2005), 21-47.  doi: 10.1016/j.jfa.2005.02.014.  Google Scholar

[21]

B. Montagnini and M. L. Demuru, Complete continuity of the free gas scattering operator in neutron thermalization theory, J. Math. Anal. Appl., 12 (1965), 49-57.  doi: 10.1016/0022-247X(65)90052-1.  Google Scholar

[22] D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1974.   Google Scholar
[23]

A. Suhadolc, Linearized Boltzmann equation in $L^1$ space, J. Math. Anal. Appl., 35 (1971), 1-13.  doi: 10.1016/0022-247X(71)90231-9.  Google Scholar

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