American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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..  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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..  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.

[2]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[3]

Joachim von Below, Gaëlle Pincet Mailly, Jean-François Rault. Growth order and blow up points for the parabolic Burgers' equation under dynamical boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 825-836. doi: 10.3934/dcdss.2013.6.825
[4]

Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126
[5]

John A. D. Appleby, Denis D. Patterson. Blow-up and superexponential growth in superlinear Volterra equations. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3993-4017. doi: 10.3934/dcds.2018174
[6]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
[7]

Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449
[8]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[9]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[10]

Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[11]

Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025
[12]

Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683
[13]

Julián López-Gómez, Pavol Quittner. Complete and energy blow-up in indefinite superlinear parabolic problems. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 169-186. doi: 10.3934/dcds.2006.14.169
[14]

Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677
[15]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[16]

Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic & Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043
[17]

Monica Marras, Stella Vernier Piro, Giuseppe Viglialoro. Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system. Conference Publications, 2015, 2015 (special) : 809-816. doi: 10.3934/proc.2015.0809
[18]

Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[19]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058
[20]

C. Brändle, F. Quirós, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure & Applied Analysis, 2005, 4 (3) : 523-536. doi: 10.3934/cpaa.2005.4.523

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (140)
  • HTML views (63)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences