Research article

Decay estimate and blow-up for a fourth order parabolic equation modeling epitaxial thin film growth

  • Received: 10 December 2022 Revised: 16 February 2023 Accepted: 27 February 2023 Published: 13 March 2023
  • MSC : 35B40, 35B44

  • In this paper, we study a fourth order parabolic equation modeling epitaxial thin film growth. By using the potential well method and some inequality techniques, we obtain the decay estimate of weak solutions. Meanwhile, the blow-up time is estimated from above and below. The blow-up rate is also derived.

    Citation: Hao Zhang, Zejian Cui, Xiangyu Xiao. Decay estimate and blow-up for a fourth order parabolic equation modeling epitaxial thin film growth[J]. AIMS Mathematics, 2023, 8(5): 11297-11311. doi: 10.3934/math.2023572

    Related Papers:

  • In this paper, we study a fourth order parabolic equation modeling epitaxial thin film growth. By using the potential well method and some inequality techniques, we obtain the decay estimate of weak solutions. Meanwhile, the blow-up time is estimated from above and below. The blow-up rate is also derived.



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    [1] B. B. King, O. Stein, M. Winkler, A fourth order parabolic equation modeling epitaxial thin film growth, J. Math. Anal. Appl., 286 (2003), 459–490. https://doi.org/10.1016/S0022-247X(03)00474-8 doi: 10.1016/S0022-247X(03)00474-8
    [2] M. Ortiz, E. Repetto, H. Si, A continuum model of kinetic roughening and coarsening in thin films, J. Mech. Phys. Solids, 47 (1999), 697–730. https://doi.org/10.1016/S0022-5096(98)00102-1 doi: 10.1016/S0022-5096(98)00102-1
    [3] C. Liu, Y. Ma, H. Tang, Lower bound of blow-up time to a fourth order parabolic equation modeling epitaxial thin film growth, Appl. Math. Lett., 111 (2021), 106609. https://doi.org/10.1016/j.aml.2020.106609 doi: 10.1016/j.aml.2020.106609
    [4] P. Górka, Logarithmic Klein-Gordon equation, Acta Phys. Pol. B, 40 (2009), 59–66.
    [5] V. Pata, S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495–1506. https://doi.org/10.1088/0951-7715/19/7/001 doi: 10.1088/0951-7715/19/7/001
    [6] H. Yang, Y. Han, Blow-up for a damped $p$-Laplacian type wave equation with logarithmic nonlinearity, J. Differ. Equ., 306 (2022), 569–589. https://doi.org/10.1016/j.jde.2021.10.036 doi: 10.1016/j.jde.2021.10.036
    [7] L. C. Nhan, L. X. Truong, Global solution and blow-up for a class of pseudo $p$-Laplacian evolution equations with logarithmic nonlinearity, Comput. Math. Appl., 73 (2017), 2076–2091. https://doi.org/10.1016/j.camwa.2017.02.030 doi: 10.1016/j.camwa.2017.02.030
    [8] H. Di, Y. Shang, Z. Song, Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Anal., 51 (2020), 102968. https://doi.org/10.1016/j.nonrwa.2019.102968 doi: 10.1016/j.nonrwa.2019.102968
    [9] H. Chen, S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equ., 258 (2015), 4424–4442. https://doi.org/10.1016/j.jde.2015.01.038 doi: 10.1016/j.jde.2015.01.038
    [10] H. Ding, J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 478 (2019), 393–420. https://doi.org/10.1016/j.jmaa.2019.05.018 doi: 10.1016/j.jmaa.2019.05.018
    [11] F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. Henri Poincar$\acute{ \text{e}}$, 23 (2006), 185–207. https://doi.org/10.1016/j.anihpc.2005.02.007 doi: 10.1016/j.anihpc.2005.02.007
    [12] K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151–177.
    [13] P. Dr$\acute{ \text{a}}$bek, S. I. Pohozaev, Positive solutions for the $p$-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A: Math., 127 (1997), 703–726. https://doi.org/10.1017/S0308210500023787 doi: 10.1017/S0308210500023787
    [14] H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equation of the form $Pu_{t} = -Au+\mathscr{F}(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371–386. https://doi.org/10.1007/BF00263041 doi: 10.1007/BF00263041
    [15] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138–146. https://doi.org/10.1137/0505015 doi: 10.1137/0505015
    [16] A. Friedman, Partial different equations, New York: Rinehart & Winston, Inc., 1969.
    [17] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, 13 (1959), 115–162.
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