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December  2020, 19(12): 5581-5590. doi: 10.3934/cpaa.2020252

Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth

Jun Zhou ,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author

Received  May 2020 Revised  July 2020 Published  December 2020 Early access  October 2020

Fund Project: The first author is supported by NSFC grant 11201380

This paper deals with a fourth order parabolic PDE arising in the theory of epitaxial growth, which was studied in [4]. We estimated the lifespan under the blow-up conditions given in [4]. Moreover, we extend the blow-up conditions of [4] from subcritical initial energy to critical initial energy.

Keywords: Epitaxial growth, forth order parabolic PDE, blow-up, lifespan.
Mathematics Subject Classification: Primary: 35J30, 35K25, 35K35; Secondary: 35K55, 35G31.
Citation: Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252
References:
[1]

K. Baghaei and M. Hesaaraki, Lower bounds for the blow-up time in the higher-dimensional nonlinear divergence form parabolic equations, C. R. Math. Acad. Sci. Paris, 351 (2013), 731-735.  doi: 10.1016/j.crma.2013.09.024.

[2] A. L. Barabási and H. E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511599798.
[3]

S. Y. Chung and M. J. Choi, A new condition for the concavity method of blow-up solutions to p-Laplacian parabolic equations, J. Differ. Equ., 265 (2018), 6384-6399.  doi: 10.1016/j.jde.2018.07.032.

[4]

C. Escudero, F. Gazzola and I. Peral, Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian, J. Math. Pures Appl., 103 (2015), 924-957.  doi: 10.1016/j.matpur.2014.09.007.

[5]

C. Escudero and I. Peral, Some fourth order nonlinear elliptic problems related to epitaxial growth, J. Differ. Equ., 254 (2013), 2515-2531.  doi: 10.1016/j.jde.2012.12.012.

[6]

C. Escudero, R. Hakl, I. Peral and and P. J. Torres, On radial stationary solutions to a model of nonequilibrium growth, Eur. J. Appl. Math., 24 (2013), 437-453.  doi: 10.1017/S0956792512000484.

[7]

Z. H. Dong and J. Zhou, Global existence and finite time blow-up for a class of thin-film equation, Z. Angew. Math. Phys., 68 (2017), Art. 89. doi: 10.1007/s00033-017-0835-3.

[8]

F. Gazzola, H. C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer, 2010. doi: 10.1007/978-3-642-12245-3.

[9]

M. Grasselli, G. Mola and A. Yagi, On the longtime behavior of solutions to a model for epitaxial growth, Osaka J. Math., 48 (2011), 987-1004. 

[10]

M. Kardar, G. Parisi and Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett., 56 (2015), 889. doi: 10.1103/PhysRevLett.56.889.

[11]

R. V. Kohn and X. D. 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]

Z. Lai and S. S Das, Kinetic growth with surface relaxation: Continuum versus atomistic models, Phys. Rev. Lett, 66 (1991), 2348. doi: 10.1103/PhysRevLett.66.2348..

[13]

H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_tt = Au + F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.1090/S0002-9947-1974-0344697-2.

[14]

B. Li and J. G. Liu, Epitaxial growth without slope selection: energetics, coarsening, and dynamic scaling, J. Nonlinear Sci., 14 (2004), 429-451.  doi: 10.1007/s00332-004-0634-9.

[15]

T. S. Lo and R. V. Kohn, A new approach to the continuum modeling of epitaxial growth: slope selection, coarsening, and the role of the uphill current, Phys. D, 161 (2002), 237-257.  doi: 10.1016/S0167-2789(01)00371-2.

[16]

M. Marsili, A. Maritan, F. Toigo and and J. R. Banavar, Stochastic growth equations and reparametrization invariance, Rev. Mod. Phys., 68 (1996), 963-983.  doi: 10.1103/RevModPhys.68.963.

[17]

L. E. Payne and G. A. Philippin, Blow-up phenomena in parabolic problems with time dependent coefficients under Dirichlet boundary conditions, Proc. Amer. Math. Soc., 141 (2013), 2309-2318.  doi: 10.1090/S0002-9939-2013-11493-0.

[18]

L. E. Payne and P. W. Schaefer, Bounds for blow-up time for the heat equation under nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1289-1296.  doi: 10.1017/S0308210511000485.

[19]

T. Sun, H. Guo and M. Grant, Dynamics of driven interfaces with a conservation law, Phys. Rev. A, 40 (1989), 6763-6766.  doi: 10.1103/physreva.40.6763.

[20]

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.

[21]

G. Y. Xu and J. Zhou, Global existence and blow-up for a fourth order parabolic equation involving the Hessian, Nonlinear Differ. Equ. Appl., 24 (2017), 41. doi: 10.1007/s00030-017-0465-7.

[22]

X. Yang and Z. F. Zhou, Improvements on lower bounds for the blow-up time under local nonlinear Neumann conditions, J. Differ. Equ., 265 (2018), 830-862.  doi: 10.1016/j.jde.2018.03.013.

[23]

J. Zhou., $L^2$-norm blow-up of solutions to a fourth order parabolic PDE involving the Hessian, J. Differ. Equ., 265 (2018), 4632-4641.  doi: 10.1016/j.jde.2018.06.015.

show all references

References:
[1]

K. Baghaei and M. Hesaaraki, Lower bounds for the blow-up time in the higher-dimensional nonlinear divergence form parabolic equations, C. R. Math. Acad. Sci. Paris, 351 (2013), 731-735.  doi: 10.1016/j.crma.2013.09.024.

[2] A. L. Barabási and H. E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511599798.
[3]

S. Y. Chung and M. J. Choi, A new condition for the concavity method of blow-up solutions to p-Laplacian parabolic equations, J. Differ. Equ., 265 (2018), 6384-6399.  doi: 10.1016/j.jde.2018.07.032.

[4]

C. Escudero, F. Gazzola and I. Peral, Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian, J. Math. Pures Appl., 103 (2015), 924-957.  doi: 10.1016/j.matpur.2014.09.007.

[5]

C. Escudero and I. Peral, Some fourth order nonlinear elliptic problems related to epitaxial growth, J. Differ. Equ., 254 (2013), 2515-2531.  doi: 10.1016/j.jde.2012.12.012.

[6]

C. Escudero, R. Hakl, I. Peral and and P. J. Torres, On radial stationary solutions to a model of nonequilibrium growth, Eur. J. Appl. Math., 24 (2013), 437-453.  doi: 10.1017/S0956792512000484.

[7]

Z. H. Dong and J. Zhou, Global existence and finite time blow-up for a class of thin-film equation, Z. Angew. Math. Phys., 68 (2017), Art. 89. doi: 10.1007/s00033-017-0835-3.

[8]

F. Gazzola, H. C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer, 2010. doi: 10.1007/978-3-642-12245-3.

[9]

M. Grasselli, G. Mola and A. Yagi, On the longtime behavior of solutions to a model for epitaxial growth, Osaka J. Math., 48 (2011), 987-1004. 

[10]

M. Kardar, G. Parisi and Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett., 56 (2015), 889. doi: 10.1103/PhysRevLett.56.889.

[11]

R. V. Kohn and X. D. 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]

Z. Lai and S. S Das, Kinetic growth with surface relaxation: Continuum versus atomistic models, Phys. Rev. Lett, 66 (1991), 2348. doi: 10.1103/PhysRevLett.66.2348..

[13]

H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_tt = Au + F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.1090/S0002-9947-1974-0344697-2.

[14]

B. Li and J. G. Liu, Epitaxial growth without slope selection: energetics, coarsening, and dynamic scaling, J. Nonlinear Sci., 14 (2004), 429-451.  doi: 10.1007/s00332-004-0634-9.

[15]

T. S. Lo and R. V. Kohn, A new approach to the continuum modeling of epitaxial growth: slope selection, coarsening, and the role of the uphill current, Phys. D, 161 (2002), 237-257.  doi: 10.1016/S0167-2789(01)00371-2.

[16]

M. Marsili, A. Maritan, F. Toigo and and J. R. Banavar, Stochastic growth equations and reparametrization invariance, Rev. Mod. Phys., 68 (1996), 963-983.  doi: 10.1103/RevModPhys.68.963.

[17]

L. E. Payne and G. A. Philippin, Blow-up phenomena in parabolic problems with time dependent coefficients under Dirichlet boundary conditions, Proc. Amer. Math. Soc., 141 (2013), 2309-2318.  doi: 10.1090/S0002-9939-2013-11493-0.

[18]

L. E. Payne and P. W. Schaefer, Bounds for blow-up time for the heat equation under nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1289-1296.  doi: 10.1017/S0308210511000485.

[19]

T. Sun, H. Guo and M. Grant, Dynamics of driven interfaces with a conservation law, Phys. Rev. A, 40 (1989), 6763-6766.  doi: 10.1103/physreva.40.6763.

[20]

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.

[21]

G. Y. Xu and J. Zhou, Global existence and blow-up for a fourth order parabolic equation involving the Hessian, Nonlinear Differ. Equ. Appl., 24 (2017), 41. doi: 10.1007/s00030-017-0465-7.

[22]

X. Yang and Z. F. Zhou, Improvements on lower bounds for the blow-up time under local nonlinear Neumann conditions, J. Differ. Equ., 265 (2018), 830-862.  doi: 10.1016/j.jde.2018.03.013.

[23]

J. Zhou., $L^2$-norm blow-up of solutions to a fourth order parabolic PDE involving the Hessian, J. Differ. Equ., 265 (2018), 4632-4641.  doi: 10.1016/j.jde.2018.06.015.

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