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Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
This paper deals with a fourth order parabolic PDE arising in the theory of epitaxial growth, which was studied in [
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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