Global existence for a thin film equation with subcritical mass

Previous Article
Optimal harvesting of a stochastic delay competitive model
DCDS-B Home
This Issue
Next Article
Saddle-node bifurcations of multiple homoclinic solutions in ODES

June 2017, 22(4): 1461-1492. doi: 10.3934/dcdsb.2017070

Global existence for a thin film equation with subcritical mass

Jian-Guo Liu ^, and Jinhuan Wang ^,

1.	School of Mathematics, Liaoning University, Shenyang 110036, China
2.	Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA

Corresponding author: Jinhuan Wang, was supported by National Natural Science Foundation of China (Grant No. 11301243) and Program for Liaoning Excellent Talents in University (Grant No. LJQ2015041)

Jian-Guo Liu was partially supported partially supported by KI-Net NSF RNMS grant No. 1107444, NSF DMS grant No. 1514826

Received December 2015 Revised November 2016 Published February 2017

In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case

$h_t+\partial_x (h^n\,\partial_{xxx} h)+\partial_x (h^{n+2}\partial_{x} h)=0,$

where

$n≥q 1$

. There exists a critical mass

$M_c=\frac{2\sqrt{6}π}{3}$

found by Witelski et al.(2004 Euro. J. of Appl. Math. 15,223-256) for

$n=1$

. We obtain global existence of a non-negative entropy weak solution if initial mass is less than

$M_c$

. For

$n≥q 4$

, entropy weak solutions are positive and unique. For

$n=1$

, a finite time blow-up occurs for solutions with initial mass larger than

$M_c$

. For the Cauchy problem with

$n=1$

and initial mass less than

$M_c$

, we show that at least one of the following long-time behavior holds:the second moment goes to infinity as the time goes to infinity or

$ h(·, t_k)\rightharpoonup 0$

in

$L^1(\mathbb{R})$

for some subsequence

${t_k} \to \infty $

.

Keywords: Long-wave instability, free-surface evolution, equilibrium, the Sz. Nagy inequality, long-time behavior.

Mathematics Subject Classification: Primary:35K65, 35K25, 39B62, 76D08.

Citation: Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070

References:

[1]	P. Álvarez-Caudevilla and V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches, Nonl. Anal., 121 (2015), 19-35. doi: 10.1016/j.na.2014.08.002. Google Scholar
[2]	E. F. Beckenbach and R. Bellman, Introduction to Inequalities Random House Inc, 1965. Google Scholar
[3]	E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation, Arch. Ration. Mech. Anal., 129 (1995), 175-200. doi: 10.1007/BF00379920. Google Scholar
[4]	F. Bernis, Finite speed of propagation and continuity of the interface for slow viscous flows, Adv. Differential Equations, 1 (1996), 337-368. Google Scholar
[5]	F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. Google Scholar
[6]	A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films, the moving contact line with a porous media cut off of Van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564. doi: 10.1088/0951-7715/7/6/002. Google Scholar
[7]	A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123. doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. Google Scholar
[8]	A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. Google Scholar
[9]	A. L. Bertozzi and M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366. doi: 10.1512/iumj.2000.49.1887. Google Scholar
[10]	M. Bertsch, L. Giacomelli and G. Karali, Thin-film equations with "partial wetting" energy: Existence of weak solutions, Physica D, 209 (2005), 17-27. doi: 10.1016/j.physd.2005.06.012. Google Scholar
[11]	M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3 (1998), 417-440. Google Scholar
[12]	S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m > 0$, Comm. Math. Phys., 323 (2013), 1017-1070. doi: 10.1007/s00220-013-1777-z. Google Scholar
[13]	M. Chugunova, M. C. Pugh and R. M. Taranets, Research Announcement: Finite-time blow up and long-wave unstable thin film equations, arXiv1008.0385v1, (2010). Google Scholar
[14]	M. Chugunova and R. M. Taranets, Blow-up with mass concentration for the long-wave unstable thin-film equation, Appl. Anal., 95 (2016), 944-962. doi: 10.1080/00036811.2015.1047829. Google Scholar
[15]	R. Dal Passo and H. Garcke, Solutions of a fourth order degenerate parabolic equation with weak initial trace, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 153-181. Google Scholar
[16]	R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170. Google Scholar
[17]	L. Giacomelli, M. V. Gnann and F. Otto, Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3, European J. Appl. Math., 24 (2013), 735-760. doi: 10.1017/S0956792513000156. Google Scholar
[18]	L. Giacomelli, H. Knüpfer and F. Otto, Smooth zero-contact-angle solutions to a thin-film equation around the steady state, J. Differential Equations, 245 (2008), 1454-1506. doi: 10.1016/j.jde.2008.06.005. Google Scholar
[19]	M. V. Gnann, Well-posedness and self-similar asymptotics for a thin-film equation, SIAM J. Math. Anal., 47 (2015), 2868--2902. doi: 10.1137/14099190X. Google Scholar
[20]	G. Grün, Droplet spreading under weak slippage: The optimal asymptotic propagation rate in the multi-dimensional case, Interfaces Free Bound., 4 (2002), 309-323. doi: 10.4171/IFB/63. Google Scholar
[21]	G. Grün, Droplet spreading under weak slippage: A basic result on nite speed of propagation, SIAM J. Math. Anal., 34 (2003), 992-1006. doi: 10.1137/S0036141002403298. Google Scholar
[22]	G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Comm. Partial Differential Equations, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193. Google Scholar
[23]	D. John, On uniqueness of weak solutions for the thin-film equation, J. Differential Equations, 259 (2015), 4122-4171. doi: 10.1016/j.jde.2015.05.013. Google Scholar
[24]	H. Knüpfer, Well-posedness for the Navier slip thin film equation in the case of partial wetting, Comm. Pure Appl. Math., 64 (2011), 1263-1296. doi: 10.1002/cpa.20376. Google Scholar
[25]	H. Knüpfer and N. Masmoudi, Darcy flow on a plate with prescribed contact angle well-posedness and lubrication approximation, Arch. Rational Mech. Anal., 218 (2015), 589-646. doi: 10.1007/s00205-015-0868-8. Google Scholar
[26]	R. S. Laugesen and M. C. Pugh, Properties of steady states for thin film equations, European J. Appl. Math., 11 (2000), 293-351. doi: 10.1017/S0956792599003794. Google Scholar
[27]	J. -L. Lions, Quelques MÃ©thodes de RÃ©solution Des ProblÃ©mes Aux Limites Non LinÃ©aires Paris, Dunod, 1969. Google Scholar
[28]	A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow Vol. 27, Cambridge University Press, 2002. Google Scholar
[29]	D. Matthes, R. J. McCann and G. SavarÃ©, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256. Google Scholar
[30]	A. Mellet, The thin film equation with non zero contact angle: A singular perturbation approach, Comm. Partial Differential Equations, 40 (2015), 1-39. doi: 10.1080/03605302.2014.895380. Google Scholar
[31]	T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X. Google Scholar
[32]	B. V. Sz. Nagy, Ãœber Integralungleichungen zwischen einer Funktion und ihrer Ableitung (German), Acta Univ. Szeged. Sect. Sci. Math., 10 (1941), 64-74. Google Scholar
[33]	F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23 (1998), 2077-2164. doi: 10.1080/03605309808821411. Google Scholar
[34]	D. SlepÄev and M. C. Pugh, Self-similar blow-up of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738. doi: 10.1512/iumj.2005.54.2569. Google Scholar
[35]	R. M. Taranets and J. R. King, On an unstable thin-film equation in multi-dimensional domains, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 105-128. doi: 10.1007/s00030-013-0240-3. Google Scholar
[36]	T. P. Witelski, A. J. xBernoff and A. L. Bertozzi, Blow-up and dissipation in a critical-case unstable thin film equation, European J. Appl. Math., 15 (2004), 223-256. doi: 10.1017/S0956792504005418. Google Scholar
[37]	Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations 2^nd edition, Singapore, World Scientific, 2001. doi: 10.1142/9789812799791. Google Scholar

show all references

References:

[1]	P. Álvarez-Caudevilla and V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches, Nonl. Anal., 121 (2015), 19-35. doi: 10.1016/j.na.2014.08.002. Google Scholar
[2]	E. F. Beckenbach and R. Bellman, Introduction to Inequalities Random House Inc, 1965. Google Scholar
[3]	E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation, Arch. Ration. Mech. Anal., 129 (1995), 175-200. doi: 10.1007/BF00379920. Google Scholar
[4]	F. Bernis, Finite speed of propagation and continuity of the interface for slow viscous flows, Adv. Differential Equations, 1 (1996), 337-368. Google Scholar
[5]	F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. Google Scholar
[6]	A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films, the moving contact line with a porous media cut off of Van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564. doi: 10.1088/0951-7715/7/6/002. Google Scholar
[7]	A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123. doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. Google Scholar
[8]	A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. Google Scholar
[9]	A. L. Bertozzi and M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366. doi: 10.1512/iumj.2000.49.1887. Google Scholar
[10]	M. Bertsch, L. Giacomelli and G. Karali, Thin-film equations with "partial wetting" energy: Existence of weak solutions, Physica D, 209 (2005), 17-27. doi: 10.1016/j.physd.2005.06.012. Google Scholar
[11]	M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3 (1998), 417-440. Google Scholar
[12]	S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m > 0$, Comm. Math. Phys., 323 (2013), 1017-1070. doi: 10.1007/s00220-013-1777-z. Google Scholar
[13]	M. Chugunova, M. C. Pugh and R. M. Taranets, Research Announcement: Finite-time blow up and long-wave unstable thin film equations, arXiv1008.0385v1, (2010). Google Scholar
[14]	M. Chugunova and R. M. Taranets, Blow-up with mass concentration for the long-wave unstable thin-film equation, Appl. Anal., 95 (2016), 944-962. doi: 10.1080/00036811.2015.1047829. Google Scholar
[15]	R. Dal Passo and H. Garcke, Solutions of a fourth order degenerate parabolic equation with weak initial trace, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 153-181. Google Scholar
[16]	R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170. Google Scholar
[17]	L. Giacomelli, M. V. Gnann and F. Otto, Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3, European J. Appl. Math., 24 (2013), 735-760. doi: 10.1017/S0956792513000156. Google Scholar
[18]	L. Giacomelli, H. Knüpfer and F. Otto, Smooth zero-contact-angle solutions to a thin-film equation around the steady state, J. Differential Equations, 245 (2008), 1454-1506. doi: 10.1016/j.jde.2008.06.005. Google Scholar
[19]	M. V. Gnann, Well-posedness and self-similar asymptotics for a thin-film equation, SIAM J. Math. Anal., 47 (2015), 2868--2902. doi: 10.1137/14099190X. Google Scholar
[20]	G. Grün, Droplet spreading under weak slippage: The optimal asymptotic propagation rate in the multi-dimensional case, Interfaces Free Bound., 4 (2002), 309-323. doi: 10.4171/IFB/63. Google Scholar
[21]	G. Grün, Droplet spreading under weak slippage: A basic result on nite speed of propagation, SIAM J. Math. Anal., 34 (2003), 992-1006. doi: 10.1137/S0036141002403298. Google Scholar
[22]	G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Comm. Partial Differential Equations, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193. Google Scholar
[23]	D. John, On uniqueness of weak solutions for the thin-film equation, J. Differential Equations, 259 (2015), 4122-4171. doi: 10.1016/j.jde.2015.05.013. Google Scholar
[24]	H. Knüpfer, Well-posedness for the Navier slip thin film equation in the case of partial wetting, Comm. Pure Appl. Math., 64 (2011), 1263-1296. doi: 10.1002/cpa.20376. Google Scholar
[25]	H. Knüpfer and N. Masmoudi, Darcy flow on a plate with prescribed contact angle well-posedness and lubrication approximation, Arch. Rational Mech. Anal., 218 (2015), 589-646. doi: 10.1007/s00205-015-0868-8. Google Scholar
[26]	R. S. Laugesen and M. C. Pugh, Properties of steady states for thin film equations, European J. Appl. Math., 11 (2000), 293-351. doi: 10.1017/S0956792599003794. Google Scholar
[27]	J. -L. Lions, Quelques MÃ©thodes de RÃ©solution Des ProblÃ©mes Aux Limites Non LinÃ©aires Paris, Dunod, 1969. Google Scholar
[28]	A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow Vol. 27, Cambridge University Press, 2002. Google Scholar
[29]	D. Matthes, R. J. McCann and G. SavarÃ©, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256. Google Scholar
[30]	A. Mellet, The thin film equation with non zero contact angle: A singular perturbation approach, Comm. Partial Differential Equations, 40 (2015), 1-39. doi: 10.1080/03605302.2014.895380. Google Scholar
[31]	T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X. Google Scholar
[32]	B. V. Sz. Nagy, Ãœber Integralungleichungen zwischen einer Funktion und ihrer Ableitung (German), Acta Univ. Szeged. Sect. Sci. Math., 10 (1941), 64-74. Google Scholar
[33]	F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23 (1998), 2077-2164. doi: 10.1080/03605309808821411. Google Scholar
[34]	D. SlepÄev and M. C. Pugh, Self-similar blow-up of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738. doi: 10.1512/iumj.2005.54.2569. Google Scholar
[35]	R. M. Taranets and J. R. King, On an unstable thin-film equation in multi-dimensional domains, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 105-128. doi: 10.1007/s00030-013-0240-3. Google Scholar
[36]	T. P. Witelski, A. J. xBernoff and A. L. Bertozzi, Blow-up and dissipation in a critical-case unstable thin film equation, European J. Appl. Math., 15 (2004), 223-256. doi: 10.1017/S0956792504005418. Google Scholar
[37]	Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations 2^nd edition, Singapore, World Scientific, 2001. doi: 10.1142/9789812799791. Google Scholar

[1]	Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020360
[2]	Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020336
[3]	Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020466
[4]	Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020 doi: 10.3934/nhm.2020033
[5]	Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
[6]	Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020403
[7]	Claude-Michel Brauner, Luca Lorenzi. Instability of free interfaces in premixed flame propagation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 575-596. doi: 10.3934/dcdss.2020363
[8]	Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021028
[9]	Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
[10]	Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002
[11]	Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122
[12]	Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328
[13]	Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299
[14]	Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
[15]	Maoli Chen, Xiao Wang, Yicheng Liu. Collision-free flocking for a time-delay system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1223-1241. doi: 10.3934/dcdsb.2020251
[16]	Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112
[17]	Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229
[18]	Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003
[19]	Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021007
[20]	Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318

2019 Impact Factor: 1.27

Tools

Metrics

Other articles
by authors

on AIMS
Jian-Guo Liu Jinhuan Wang
on Google Scholar
Jian-Guo Liu Jinhuan Wang

2019 Impact Factor: 1.27

Tools

Article outline

2019 Impact Factor: 1.27

Tools

Metrics

Other articles
by authors

on AIMS
Jian-Guo Liu Jinhuan Wang
on Google Scholar
Jian-Guo Liu Jinhuan Wang

[Back to Top]