A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films

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December 2021, 14(12): 4367-4381. doi: 10.3934/dcdss.2021112

A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films

Yang Liu ^1, and Wenke Li ^2,,

1.	College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China
2.	College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China

^* Corresponding author: liwenke@hrbeu.edu.cn

Received July 2021 Revised August 2021 Published December 2021 Early access October 2021

Fund Project: This work is supported by the Science and Technology Plan Project of Gansu Province in China (Grant No. 21JR1RA200), the Talent Introduction Research Project of Northwest Minzu University (Grant No. xbmuyjrc2021008), the Innovation Team Project of Northwest Minzu University (Grant No. 1110130131), the First-Rate Discipline of Northwest Minzu University (Grant No. 2019XJYLZY-02) and the National Natural Science Foundation of China (Grant No. 12102100)

In this paper, the initial-boundary value problem for a class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films is studied. By means of the theory of potential wells, the global existence, asymptotic behavior and finite time blow-up of weak solutions are obtained.

Keywords: Fourth-order nonlinear parabolic equations, existence, asymptotic behavior, blow-up.

Mathematics Subject Classification: Primary: 35K35, 35A01, 35B40, 35B44.

Citation: Yang Liu, Wenke Li. A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4367-4381. doi: 10.3934/dcdss.2021112

References:

[1]	L. Agélas, Global regularity of solutions of equation modeling epitaxy thin film growth in $\mathbb{R}^d$, $d = 1, 2$, J. Evol. Equ., 15 (2015), 89-106. doi: 10.1007/s00028-014-0250-6.
[2]	D. Blömker and C. Gugg, On the existence of solutions for amorphous molecular beam epitaxy, Nonlinear Anal. Real World Appl., 3 (2002), 61-73. doi: 10.1016/S1468-1218(01)00013-X.
[3]	D. Blömker, C. Gugg and M. Raible, Thin-film-growth models: Roughness and correlation functions, Eur. J. Appl. Math., 13 (2002), 385-402. doi: 10.1017/S0956792502004886.
[4]	M. Capiński and D. Gatarek, Stochastic equations in Hilbert space with applications to Navier-Stokes equation in any dimensions, J. Functional Anal., 126 (1994), 26-35. doi: 10.1006/jfan.1994.1140.
[5]	H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203. doi: 10.3934/dcds.2019051.
[6]	S. Das Sarma and S. V. Ghaisas, Solid-on-solid rules and models for nonequilibrium growth in $2+1$ dimensions, Phys. Rev. Lett., 69 (1992), 3762-3765.
[7]	M. Dimova, N. Kolkovska and N. Kutev, Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy, Elec. Res. Arch., 28 (2020), 671-689. doi: 10.3934/era.2020035.
[8]	J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Elec. Res. Arch., 28 (2020), 347-367. doi: 10.3934/era.2020020.
[9]	J. M. Kim and S. Das Sarma, Discrete models for conserved growth equations, Phys. Rev. Lett., 72 (1994), 2903-2906. doi: 10.1103/PhysRevLett.72.2903.
[10]	B. B. King, O. Stein and M. Winkler, A fourth-order parabolic equation modeling epitaxial thin film growth, J. Math. Anal. Appl., 286 (2003), 459-490. doi: 10.1016/S0022-247X(03)00474-8.
[11]	R. V. Kohn and X. Yan, Upper bound on the coarsening rate for an epitaxial growth model, Comm. Pure Appl. Math., 56 (2003), 1549-1564. doi: 10.1002/cpa.10103.
[12]	W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632. doi: 10.1515/anona-2020-0016.
[13]	M. Liao, Q. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional $p$-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591. doi: 10.1515/anona-2020-0066.
[14]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.
[15]	W. Liu, Z. Chen and Z. Tu, New general decay result for a fourth-order Moore-Gibson-Thompson equation with memory, Elec. Res. Arch., 28 (2020), 433-457. doi: 10.3934/era.2020025.
[16]	Y. Liu, Long-time behavior of a class of viscoelastic plate equations, Elec. Res. Arch., 28 (2020), 311-326. doi: 10.3934/era.2020018.
[17]	Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687. doi: 10.1016/j.na.2005.09.011.
[18]	W. W. Mullins, Theory of thermal grooving, J. Appl. Phys., 28 (1957), 333-339. doi: 10.1063/1.1722742.
[19]	T. Niimura, Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations, Discrete Contin. Dyn. Syst., 40 (2020), 2561-2591. doi: 10.3934/dcds.2020141.
[20]	M. Ortiz, E. A. Repetto and H. Si, A continuum model of kinetic roughening and coarsening in thin films, J. Mech. Phys. Solids, 47 (1999), 697-730. doi: 10.1016/S0022-5096(98)00102-1.
[21]	L. E. Payne and D. H. Sattinger, Sadle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. doi: 10.1007/BF02761595.
[22]	T. P. Schulze and R. V. Kohn, A geometric model for coarsening during spiral-mode growth of thin films, Phys. D, 132 (1999), 520-542. doi: 10.1016/S0167-2789(99)00108-6.
[23]	O. Stein and M. Winkler, Amorphous molecular beam epitaxy: Global solutions and absorbing sets, Eur. J. Appl. Math., 16 (2005), 767-798. doi: 10.1017/S0956792505006315.
[24]	X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288. doi: 10.1515/anona-2020-0141.
[25]	M. Winkler, Global solutions in higher dimensions to a fourth order parabolic equation modeling epitaxial thin film growth, Z. Angew. Math. Phys., 62 (2011), 575-608. doi: 10.1007/s00033-011-0128-1.
[26]	R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763. doi: 10.1016/j.jfa.2013.03.010.
[27]	R. Xu, M. Zhang, S. Chen, Y. Yang and J. Shen, The initial-boundary value problems for a class of six order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649. doi: 10.3934/dcds.2017244.
[28]	X.-G. Yang, M. J. D. Nascimento and M. L. Pelicer, Uniform attractors for non-autonomous plate equations with $p$-Laplacian perturbation and critical nonlinearities, Discrete Contin. Dyn. Syst., 40 (2020), 1937-1961. doi: 10.3934/dcds.2020100.
[29]	Z. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540.
[30]	A. Zangwill, Some causes and a consequence of epitaxial roughening, J. Crystal Growth, 163 (1996), 8-21. doi: 10.1016/0022-0248(95)01048-3.
[31]	M. Zhang and M. S. Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal., 9 (2020), 882-894. doi: 10.1515/anona-2020-0031.
[32]	X. Zhao and C. Liu, Time-periodic solution of a 2D fourth-order nonlinear parabolic equation, Proc. Indian Acad. Sci. (Math. Sci.), 124 (2014), 349-364. doi: 10.1007/s12044-014-0180-9.

show all references

References:

[1]	L. Agélas, Global regularity of solutions of equation modeling epitaxy thin film growth in $\mathbb{R}^d$, $d = 1, 2$, J. Evol. Equ., 15 (2015), 89-106. doi: 10.1007/s00028-014-0250-6.
[2]	D. Blömker and C. Gugg, On the existence of solutions for amorphous molecular beam epitaxy, Nonlinear Anal. Real World Appl., 3 (2002), 61-73. doi: 10.1016/S1468-1218(01)00013-X.
[3]	D. Blömker, C. Gugg and M. Raible, Thin-film-growth models: Roughness and correlation functions, Eur. J. Appl. Math., 13 (2002), 385-402. doi: 10.1017/S0956792502004886.
[4]	M. Capiński and D. Gatarek, Stochastic equations in Hilbert space with applications to Navier-Stokes equation in any dimensions, J. Functional Anal., 126 (1994), 26-35. doi: 10.1006/jfan.1994.1140.
[5]	H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203. doi: 10.3934/dcds.2019051.
[6]	S. Das Sarma and S. V. Ghaisas, Solid-on-solid rules and models for nonequilibrium growth in $2+1$ dimensions, Phys. Rev. Lett., 69 (1992), 3762-3765.
[7]	M. Dimova, N. Kolkovska and N. Kutev, Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy, Elec. Res. Arch., 28 (2020), 671-689. doi: 10.3934/era.2020035.
[8]	J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Elec. Res. Arch., 28 (2020), 347-367. doi: 10.3934/era.2020020.
[9]	J. M. Kim and S. Das Sarma, Discrete models for conserved growth equations, Phys. Rev. Lett., 72 (1994), 2903-2906. doi: 10.1103/PhysRevLett.72.2903.
[10]	B. B. King, O. Stein and M. Winkler, A fourth-order parabolic equation modeling epitaxial thin film growth, J. Math. Anal. Appl., 286 (2003), 459-490. doi: 10.1016/S0022-247X(03)00474-8.
[11]	R. V. Kohn and X. Yan, Upper bound on the coarsening rate for an epitaxial growth model, Comm. Pure Appl. Math., 56 (2003), 1549-1564. doi: 10.1002/cpa.10103.
[12]	W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632. doi: 10.1515/anona-2020-0016.
[13]	M. Liao, Q. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional $p$-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591. doi: 10.1515/anona-2020-0066.
[14]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.
[15]	W. Liu, Z. Chen and Z. Tu, New general decay result for a fourth-order Moore-Gibson-Thompson equation with memory, Elec. Res. Arch., 28 (2020), 433-457. doi: 10.3934/era.2020025.
[16]	Y. Liu, Long-time behavior of a class of viscoelastic plate equations, Elec. Res. Arch., 28 (2020), 311-326. doi: 10.3934/era.2020018.
[17]	Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687. doi: 10.1016/j.na.2005.09.011.
[18]	W. W. Mullins, Theory of thermal grooving, J. Appl. Phys., 28 (1957), 333-339. doi: 10.1063/1.1722742.
[19]	T. Niimura, Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations, Discrete Contin. Dyn. Syst., 40 (2020), 2561-2591. doi: 10.3934/dcds.2020141.
[20]	M. Ortiz, E. A. Repetto and H. Si, A continuum model of kinetic roughening and coarsening in thin films, J. Mech. Phys. Solids, 47 (1999), 697-730. doi: 10.1016/S0022-5096(98)00102-1.
[21]	L. E. Payne and D. H. Sattinger, Sadle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. doi: 10.1007/BF02761595.
[22]	T. P. Schulze and R. V. Kohn, A geometric model for coarsening during spiral-mode growth of thin films, Phys. D, 132 (1999), 520-542. doi: 10.1016/S0167-2789(99)00108-6.
[23]	O. Stein and M. Winkler, Amorphous molecular beam epitaxy: Global solutions and absorbing sets, Eur. J. Appl. Math., 16 (2005), 767-798. doi: 10.1017/S0956792505006315.
[24]	X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288. doi: 10.1515/anona-2020-0141.
[25]	M. Winkler, Global solutions in higher dimensions to a fourth order parabolic equation modeling epitaxial thin film growth, Z. Angew. Math. Phys., 62 (2011), 575-608. doi: 10.1007/s00033-011-0128-1.
[26]	R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763. doi: 10.1016/j.jfa.2013.03.010.
[27]	R. Xu, M. Zhang, S. Chen, Y. Yang and J. Shen, The initial-boundary value problems for a class of six order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649. doi: 10.3934/dcds.2017244.
[28]	X.-G. Yang, M. J. D. Nascimento and M. L. Pelicer, Uniform attractors for non-autonomous plate equations with $p$-Laplacian perturbation and critical nonlinearities, Discrete Contin. Dyn. Syst., 40 (2020), 1937-1961. doi: 10.3934/dcds.2020100.
[29]	Z. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540.
[30]	A. Zangwill, Some causes and a consequence of epitaxial roughening, J. Crystal Growth, 163 (1996), 8-21. doi: 10.1016/0022-0248(95)01048-3.
[31]	M. Zhang and M. S. Ahmed, Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential, Adv. Nonlinear Anal., 9 (2020), 882-894. doi: 10.1515/anona-2020-0031.
[32]	X. Zhao and C. Liu, Time-periodic solution of a 2D fourth-order nonlinear parabolic equation, Proc. Indian Acad. Sci. (Math. Sci.), 124 (2014), 349-364. doi: 10.1007/s12044-014-0180-9.

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