doi: 10.3934/dcdss.2021112
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A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films

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 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.

Citation: Yang Liu, Wenke Li. A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films. Discrete & Continuous Dynamical Systems - S, 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.  Google Scholar

[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.  Google Scholar

[3]

D. BlömkerC. Gugg and M. Raible, Thin-film-growth models: Roughness and correlation functions, Eur. J. Appl. Math., 13 (2002), 385-402.  doi: 10.1017/S0956792502004886.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

M. DimovaN. 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.  Google Scholar

[8]

J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Elec. Res. Arch., 28 (2020), 347-367.  doi: 10.3934/era.2020020.  Google Scholar

[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.  Google Scholar

[10]

B. B. KingO. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. LiaoQ. 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.  Google Scholar

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J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar

[15]

W. LiuZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

W. W. Mullins, Theory of thermal grooving, J. Appl. Phys., 28 (1957), 333-339.  doi: 10.1063/1.1722742.  Google Scholar

[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.  Google Scholar

[20]

M. OrtizE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

R. XuM. ZhangS. ChenY. 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.  Google Scholar

[28]

X.-G. YangM. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

D. BlömkerC. Gugg and M. Raible, Thin-film-growth models: Roughness and correlation functions, Eur. J. Appl. Math., 13 (2002), 385-402.  doi: 10.1017/S0956792502004886.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

M. DimovaN. 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.  Google Scholar

[8]

J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Elec. Res. Arch., 28 (2020), 347-367.  doi: 10.3934/era.2020020.  Google Scholar

[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.  Google Scholar

[10]

B. B. KingO. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. LiaoQ. 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.  Google Scholar

[14]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.  Google Scholar

[15]

W. LiuZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

W. W. Mullins, Theory of thermal grooving, J. Appl. Phys., 28 (1957), 333-339.  doi: 10.1063/1.1722742.  Google Scholar

[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.  Google Scholar

[20]

M. OrtizE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

R. XuM. ZhangS. ChenY. 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.  Google Scholar

[28]

X.-G. YangM. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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