doi: 10.3934/ipi.2022024
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Well-posedness of an inverse problem for two- and three-dimensional convective Brinkman-Forchheimer equations with the final overdetermination

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, India

*Corresponding author: Manil T. Mohan

Received  August 2021 Revised  April 2022 Early access May 2022

Fund Project: The authors are supported by INSPIRE Faculty Award Grant IFA17-MA110

In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations:
$ \begin{align*} \boldsymbol{u}_t-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p = \boldsymbol{F}: = \boldsymbol{f} g, \ \ \ \nabla\cdot\boldsymbol{u} = 0, \end{align*} $
in bounded domains
$ \Omega\subset\mathbb{R}^d $
(
$ d = 2, 3 $
) with smooth boundary, where
$ \alpha, \beta, \mu>0 $
and
$ r\in[1, \infty) $
. The CBF equations describe the motion of incompressible fluid flows in a saturated porous medium. The inverse problem under our consideration consists of reconstructing the vector-valued velocity function
$ \boldsymbol{u} $
, the pressure gradient
$ \nabla p $
and the vector-valued function
$ \boldsymbol{f} $
. We prove the well-posedness result (existence, uniqueness and stability) of an inverse problem for 2D and 3D CBF equations with the final overdetermination condition using Schauder's fixed point theorem for arbitrary smooth initial data. The well-posedness results hold for
$ r\geq 1 $
in two dimensions and for
$ r \geq 3 $
in three dimensions. The global solvability results available in the literature helped us to obtain the uniqueness and stability results for the model with fast growing nonlinearities.
Citation: Pardeep Kumar, Manil T. Mohan. Well-posedness of an inverse problem for two- and three-dimensional convective Brinkman-Forchheimer equations with the final overdetermination. Inverse Problems and Imaging, doi: 10.3934/ipi.2022024
References:
[1]

S. N. Antontsev and H. B. de Oliveira, The Navier-Stokes problem modified by an absorption term, Appl. Anal., 89 (2010), 1805-1825.  doi: 10.1080/00036811.2010.495341.

[2]

M. BadraF. Caubet and J. Dardé, Stability estimates for Navier-Stokes equations and application to inverse problems, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2379-2407.  doi: 10.3934/dcdsb.2016052.

[3] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993. 
[4]

V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, J. Math. Anal. Appl., 255 (2001), 281-307.  doi: 10.1006/jmaa.2000.7256.

[5]

I. Bushuyev, Global uniqueness for inverse parabolic problems with final observation, Inverse Problems, 11 (1995), 11-16.  doi: 10.1088/0266-5611/11/4/001.

[6]

X. Cai and Q. Jiu, Weak and Strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.

[7]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Mat. Sem. Univ. Padova, 31 (1961), 308-340. 

[8]

A. Y. Chebotarev, Inverse problem for Navier-Stokes systems with finite-dimensional overdetermination, Differ. Uravn., 48 (2012), 1153-1160.  doi: 10.1134/S0012266112080101.

[9]

A. Y. Chebotarev, Inverse problems for stationary Navier-Stokes systems, Comput. Math. Math. Phys., 54 (2014), 537-545.  doi: 10.1134/S0965542514030038.

[10]

M. ChouliO. Y. ImanuvilovJ.-P. Puel and M. Yamamoto, Inverse source problem for linearized Navier-Stokes equations with data in arbitrary sub-domain, Appl. Anal., 92 (2013), 2127-2143.  doi: 10.1080/00036811.2012.718334.

[11]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM Philadelphia, 2013.

[12]

J. FanM. Di CristoY. Jiang and G. Nakamura, Inverse viscosity problem for the Navier-Stokes equation, J. Math. Anal. Appl., 365 (2010), 750-757.  doi: 10.1016/j.jmaa.2009.12.012.

[13]

J. Fan and G. Nakamura, Local solvability of an inverse problem to the density-dependent Navier-Stokes equations, Appl. Anal., 87 (2008), 1255-1265.  doi: 10.1080/00036810802428920.

[14]

J. Fan and G. Nakamura, Well-posedness of an inverse problem of Navier-Stokes equations with the final overdetermination, J. Inverse Ill-Posed Probl., 17 (2009), 565-584.  doi: 10.1515/JIIP.2009.035.

[15]

C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous Approximation in Lebesgue and Sobolev Norms Via Eigenspaces, https://arXiv.org/abs/1904.03337.

[16]

D. Fujiwara and H. Morimoto, An $L^r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700. 

[17]

N. L. Gol'dman, Determination of the right-hand side in a quasilinear parobolic equation with a terminal observation, Diff. Eqns., 41 (2005), 384-392. 

[18]

K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, J. Differential Equations, 263 (2017), 7141-7161.  doi: 10.1016/j.jde.2017.08.001.

[19]

K. W. Hajduk, J. C. Robinson and W. Sadowski, Robustness of regularity for the 3D convective Brinkman-Forchheimer equations, J. Math. Anal. Appl., 500 (2021), 125058, 23 pp. doi: 10.1016/j.jmaa.2021.125058.

[20]

O. Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions, Inverse Problems, 31 (2015), 035004, 46 pp. doi: 10.1088/0266-5611/31/3/035004.

[21]

V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185-209.  doi: 10.1002/cpa.3160440203.

[22]

V. Isakov, Inverse Problems for Partial Differential Equation, 2$^{ed}$ edition, Now York, Springer, 2006.

[23]

Y. Jiang, J. Fan, S. Nagayasu and G. Nakamura, Local solvability of an inverse problem to the Navier-Stokes equation with memory term, Inverse Problems, 36 (2020), 065007, 14 pp. doi: 10.1088/1361-6420/ab7e05.

[24]

V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054.  doi: 10.3934/cpaa.2012.11.2037.

[25]

A. I. Korotkii, Inverse problems of reconstructing parameters of the Navier-Stokes system, J. Math. Sci., 140 (2007), 808-831.  doi: 10.1007/s10958-007-0019-3.

[26]

P. Kumar, K. Kinra and M. T. Mohan, A local in time existence and uniqueness result of an inverse problem for the Kelvin-Voigt fluids, Inverse Problems, 37 (2021), 085005, 34 pp. doi: 10.1088/1361-6420/ac1050.

[27]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

[28]

R. Y. LaiG. Uhlmann and J. N. Wang, Inverse boundary value problem for the Stokes and the Navier-Stokes equations in the plane, Arch. Ration. Mech. Anal., 215 (2015), 811-829.  doi: 10.1007/s00205-014-0794-1.

[29]

P. A. MarkowichE. S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328.  doi: 10.1088/0951-7715/29/4/1292.

[30]

M. T. Mohan, On the convective Brinkman-Forchheimer equations, (submitted).

[31]

M. T. Mohan, Stochastic convective Brinkman-Forchheimer equations (submitted), https://arXiv.org/abs/2007.09376.

[32]

A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000.

[33]

A. I. Prilepko and D. S. Tkachenko, Well-posedness of the inverse source problem for parabolic systems, Diff. Eqns., {40} (2004), 1619–1626. doi: 10.1007/s10625-005-0080-y.

[34] J. C. RobinsonJ. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2016.  doi: 10.1017/CBO9781139095143.
[35]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.

[36]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.

[37]

I. A. Vasin and A. I. Prilepko, The solvability of three dimensional inverse problem for the nonlinear Navier-Stokes equations, U. S. S. R. Comput. Math. and Math. Phys., 30 (1990), 189-199.  doi: 10.1016/0041-5553(90)90177-T.

[38]

Z. ZhangX. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419.  doi: 10.1016/j.jmaa.2010.11.019.

[39]

Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825.  doi: 10.1016/j.aml.2012.02.029.

[40]

H. Zou, et al, Handbook of Differential Equations: Stationary Partial Differential Equations, Volume VI, Elsevier.

show all references

References:
[1]

S. N. Antontsev and H. B. de Oliveira, The Navier-Stokes problem modified by an absorption term, Appl. Anal., 89 (2010), 1805-1825.  doi: 10.1080/00036811.2010.495341.

[2]

M. BadraF. Caubet and J. Dardé, Stability estimates for Navier-Stokes equations and application to inverse problems, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2379-2407.  doi: 10.3934/dcdsb.2016052.

[3] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993. 
[4]

V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, J. Math. Anal. Appl., 255 (2001), 281-307.  doi: 10.1006/jmaa.2000.7256.

[5]

I. Bushuyev, Global uniqueness for inverse parabolic problems with final observation, Inverse Problems, 11 (1995), 11-16.  doi: 10.1088/0266-5611/11/4/001.

[6]

X. Cai and Q. Jiu, Weak and Strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.

[7]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Mat. Sem. Univ. Padova, 31 (1961), 308-340. 

[8]

A. Y. Chebotarev, Inverse problem for Navier-Stokes systems with finite-dimensional overdetermination, Differ. Uravn., 48 (2012), 1153-1160.  doi: 10.1134/S0012266112080101.

[9]

A. Y. Chebotarev, Inverse problems for stationary Navier-Stokes systems, Comput. Math. Math. Phys., 54 (2014), 537-545.  doi: 10.1134/S0965542514030038.

[10]

M. ChouliO. Y. ImanuvilovJ.-P. Puel and M. Yamamoto, Inverse source problem for linearized Navier-Stokes equations with data in arbitrary sub-domain, Appl. Anal., 92 (2013), 2127-2143.  doi: 10.1080/00036811.2012.718334.

[11]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM Philadelphia, 2013.

[12]

J. FanM. Di CristoY. Jiang and G. Nakamura, Inverse viscosity problem for the Navier-Stokes equation, J. Math. Anal. Appl., 365 (2010), 750-757.  doi: 10.1016/j.jmaa.2009.12.012.

[13]

J. Fan and G. Nakamura, Local solvability of an inverse problem to the density-dependent Navier-Stokes equations, Appl. Anal., 87 (2008), 1255-1265.  doi: 10.1080/00036810802428920.

[14]

J. Fan and G. Nakamura, Well-posedness of an inverse problem of Navier-Stokes equations with the final overdetermination, J. Inverse Ill-Posed Probl., 17 (2009), 565-584.  doi: 10.1515/JIIP.2009.035.

[15]

C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous Approximation in Lebesgue and Sobolev Norms Via Eigenspaces, https://arXiv.org/abs/1904.03337.

[16]

D. Fujiwara and H. Morimoto, An $L^r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700. 

[17]

N. L. Gol'dman, Determination of the right-hand side in a quasilinear parobolic equation with a terminal observation, Diff. Eqns., 41 (2005), 384-392. 

[18]

K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, J. Differential Equations, 263 (2017), 7141-7161.  doi: 10.1016/j.jde.2017.08.001.

[19]

K. W. Hajduk, J. C. Robinson and W. Sadowski, Robustness of regularity for the 3D convective Brinkman-Forchheimer equations, J. Math. Anal. Appl., 500 (2021), 125058, 23 pp. doi: 10.1016/j.jmaa.2021.125058.

[20]

O. Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions, Inverse Problems, 31 (2015), 035004, 46 pp. doi: 10.1088/0266-5611/31/3/035004.

[21]

V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185-209.  doi: 10.1002/cpa.3160440203.

[22]

V. Isakov, Inverse Problems for Partial Differential Equation, 2$^{ed}$ edition, Now York, Springer, 2006.

[23]

Y. Jiang, J. Fan, S. Nagayasu and G. Nakamura, Local solvability of an inverse problem to the Navier-Stokes equation with memory term, Inverse Problems, 36 (2020), 065007, 14 pp. doi: 10.1088/1361-6420/ab7e05.

[24]

V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054.  doi: 10.3934/cpaa.2012.11.2037.

[25]

A. I. Korotkii, Inverse problems of reconstructing parameters of the Navier-Stokes system, J. Math. Sci., 140 (2007), 808-831.  doi: 10.1007/s10958-007-0019-3.

[26]

P. Kumar, K. Kinra and M. T. Mohan, A local in time existence and uniqueness result of an inverse problem for the Kelvin-Voigt fluids, Inverse Problems, 37 (2021), 085005, 34 pp. doi: 10.1088/1361-6420/ac1050.

[27]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

[28]

R. Y. LaiG. Uhlmann and J. N. Wang, Inverse boundary value problem for the Stokes and the Navier-Stokes equations in the plane, Arch. Ration. Mech. Anal., 215 (2015), 811-829.  doi: 10.1007/s00205-014-0794-1.

[29]

P. A. MarkowichE. S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328.  doi: 10.1088/0951-7715/29/4/1292.

[30]

M. T. Mohan, On the convective Brinkman-Forchheimer equations, (submitted).

[31]

M. T. Mohan, Stochastic convective Brinkman-Forchheimer equations (submitted), https://arXiv.org/abs/2007.09376.

[32]

A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000.

[33]

A. I. Prilepko and D. S. Tkachenko, Well-posedness of the inverse source problem for parabolic systems, Diff. Eqns., {40} (2004), 1619–1626. doi: 10.1007/s10625-005-0080-y.

[34] J. C. RobinsonJ. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2016.  doi: 10.1017/CBO9781139095143.
[35]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.

[36]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.

[37]

I. A. Vasin and A. I. Prilepko, The solvability of three dimensional inverse problem for the nonlinear Navier-Stokes equations, U. S. S. R. Comput. Math. and Math. Phys., 30 (1990), 189-199.  doi: 10.1016/0041-5553(90)90177-T.

[38]

Z. ZhangX. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419.  doi: 10.1016/j.jmaa.2010.11.019.

[39]

Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825.  doi: 10.1016/j.aml.2012.02.029.

[40]

H. Zou, et al, Handbook of Differential Equations: Stationary Partial Differential Equations, Volume VI, Elsevier.

[1]

Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control and Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509

[2]

Manil T. Mohan. Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations. Evolution Equations and Control Theory, 2022, 11 (3) : 649-679. doi: 10.3934/eect.2021020

[3]

Kush Kinra, Manil T. Mohan. Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021061

[4]

Varga K. Kalantarov, Sergey Zelik. Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2037-2054. doi: 10.3934/cpaa.2012.11.2037

[5]

Yuncheng You, Caidi Zhao, Shengfan Zhou. The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3787-3800. doi: 10.3934/dcds.2012.32.3787

[6]

Timir Karmakar, Meraj Alam, G. P. Raja Sekhar. Analysis of Brinkman-Forchheimer extended Darcy's model in a fluid saturated anisotropic porous channel. Communications on Pure and Applied Analysis, 2022, 21 (3) : 845-865. doi: 10.3934/cpaa.2022001

[7]

Qiangheng Zhang, Yangrong Li. Regular attractors of asymptotically autonomous stochastic 3D Brinkman-Forchheimer equations with delays. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3515-3537. doi: 10.3934/cpaa.2021117

[8]

Shu Wang, Mengmeng Si, Rong Yang. Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1621-1636. doi: 10.3934/cpaa.2022034

[9]

Jaan Janno, Kairi Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems and Imaging, 2009, 3 (1) : 17-41. doi: 10.3934/ipi.2009.3.17

[10]

Wenjing Liu, Rong Yang, Xin-Guang Yang. Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1907-1930. doi: 10.3934/cpaa.2021052

[11]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[12]

Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4051-4059. doi: 10.3934/dcds.2022045

[13]

Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775

[14]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692

[15]

Adriana C. Briozzo, María F. Natale, Domingo A. Tarzia. The Stefan problem with temperature-dependent thermal conductivity and a convective term with a convective condition at the fixed face. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1209-1220. doi: 10.3934/cpaa.2010.9.1209

[16]

Manil T. Mohan. Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3393-3436. doi: 10.3934/dcdsb.2020067

[17]

Ruhua Wang, Senjian An, Wanquan Liu, Ling Li. Fixed-point algorithms for inverse of residual rectifier neural networks. Mathematical Foundations of Computing, 2021, 4 (1) : 31-44. doi: 10.3934/mfc.2020024

[18]

Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 397-415. doi: 10.3934/ipi.2021055

[19]

El Mustapha Ait Ben Hassi, Salah-Eddine Chorfi, Lahcen Maniar, Omar Oukdach. Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions. Evolution Equations and Control Theory, 2021, 10 (4) : 837-859. doi: 10.3934/eect.2020094

[20]

Abdelaaziz Sbai, Youssef El Hadfi, Mohammed Srati, Noureddine Aboutabit. Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 213-227. doi: 10.3934/dcdss.2021015

2021 Impact Factor: 1.483

Article outline

[Back to Top]