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Pattern formation in the doubly-nonlocal Fisher-KPP equation
Global existence and decay to equilibrium for some crystal surface models
1. | Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria. Avda. Los Castros s/n, Santander, Spain |
2. | Dipartimento di Matematica, Università degli Studi Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy |
$ \ \ \ \ \ {\partial _t} u = \Delta e^{-\Delta u}, \\ {\partial _t} u = -u^2\Delta ^2(u^3). $ |
References:
[1] |
D. Ambrose, The radius of analyticity for solutions to a problem in epitaxial growth on the torus, arXiv preprint, arXiv: 1807.01740, 2018. |
[2] |
H. Bae, R. Granero-Belinchón and O. Lazar,
Global existence of weak solutions to dissipative transport equations with nonlocal velocity, Nonlinearity, 31 (2018), 1484-1515.
doi: 10.1088/1361-6544/aaa2e0. |
[3] |
G. Bruell and R. Granero-Belinchón, On the thin film Muskat and the thin film Stokes equations, arXiv preprint, 2018, arXiv: 1802.05509 [math.AP]. |
[4] |
J. Burczak and R. Granero-Belinchón,
On a generalized doubly parabolic keller–segel system in one spatial dimension, Mathematical Models and Methods in Applied Sciences, 26 (2016), 111-160.
doi: 10.1142/S0218202516500044. |
[5] |
P. Constantin, D. Córdoba, F. Gancedo, L. Rodriguez-Piazza and R. M. Strain,
On the muskat problem: Global in time results in 2d and 3d, American Journal of Mathematics, 138 (2016), 1455-1494.
doi: 10.1353/ajm.2016.0044. |
[6] |
D. Córdoba and F. Gancedo,
Contour dynamics of incompressible 3-d fluids in a porous medium with different densities, Communications in Mathematical Physics, 273 (2007), 445-471.
doi: 10.1007/s00220-007-0246-y. |
[7] |
F. Gancedo, E. Garcia-Juarez, N. Patel and R. M. Strain, On the muskat problem with viscosity jump: Global in time results, arXiv preprint, 2017, arXiv: 1710.11604. |
[8] |
Y. Gao, H. Ji, J.-G. Liu and T. P. Witelski,
A vicinal surface model for epitaxial growth with logarithmic free energy, AIMS, 23 (2018), 4433-4453.
doi: 10.3934/dcdsb.2018170. |
[9] |
Y. Gao, J.-G. Liu and J. Lu,
Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime, SIAM Journal on Mathematical Analysis, 49 (2017), 1705-1731.
doi: 10.1137/16M1094543. |
[10] |
Y. Gao, J.-G. Liu and X. Y. Lu, Gradient Flow Approach to an Exponential Thin Film Equation: Global Existence and Latent Singularity, ESAIM: COCV, Forthcoming article, 2018, arXiv: 1710.06995.
doi: 10.1051/cocv/2018037. |
[11] |
Y. Giga and R. V. Kohn,
Scale-invariant extinction time estimates for some singular diffusion equations, Discrete Contin. Dyn. Syst., 30 (2011), 509-535.
doi: 10.3934/dcds.2011.30.509. |
[12] |
J. Krug, H. T. Dobbs and S. Majaniemi, Adatom mobility for the solid-on-solid model, Zeitschrift für Physik B Condensed Matter, 97 (1995), 281–291.
doi: 10.1007/BF01307478. |
[13] |
J.-G. Liu and X. Xu,
Existence theorems for a multidimensional crystal surface model, SIAM Journal on Mathematical Analysis, 48 (2016), 3667-3687.
doi: 10.1137/16M1059400. |
[14] |
J. G. Liu and X. Xu,
Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions, SIAM Journal on Mathematical Analysis, 49 (2017), 2220-2245.
doi: 10.1137/16M1098474. |
[15] |
Jian-Guo Liu and Robert M. Strain., Global stability for solutions to the exponential PDE describing epitaxial growth., arXiv preprint arXiv: 1805.02246, 2018. |
[16] |
J. L. Marzuola and J. Weare, Relaxation of a family of broken-bond crystal-surface models, Physical Review E, 88 (2013), 032403.
doi: 10.1103/PhysRevE.88.032403. |
[17] |
N. Patel and R. M. Strain,
Large time decay estimates for the muskat equation, Communications in Partial Differential Equations, 42 (2017), 977-999.
doi: 10.1080/03605302.2017.1321661. |
[18] |
H. A. H. Shehadeh, R. V. Kohn and J. Weare,
The evolution of a crystal surface: Analysis of a one-dimensional step train connecting two facets in the adl regime, Physica D: Nonlinear Phenomena, 240 (2011), 1771-1784.
doi: 10.1016/j.physd.2011.07.016. |
[19] |
J. Simon,
Compact sets in the space $L^{p}(O, T; B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 65-96.
doi: 10.1007/BF01762360. |
[20] |
X. Xu, Existence theorems for a crystal surface model involving the p-laplacian operator, arXiv preprint, 2017, arXiv: 1711.07405. |
show all references
References:
[1] |
D. Ambrose, The radius of analyticity for solutions to a problem in epitaxial growth on the torus, arXiv preprint, arXiv: 1807.01740, 2018. |
[2] |
H. Bae, R. Granero-Belinchón and O. Lazar,
Global existence of weak solutions to dissipative transport equations with nonlocal velocity, Nonlinearity, 31 (2018), 1484-1515.
doi: 10.1088/1361-6544/aaa2e0. |
[3] |
G. Bruell and R. Granero-Belinchón, On the thin film Muskat and the thin film Stokes equations, arXiv preprint, 2018, arXiv: 1802.05509 [math.AP]. |
[4] |
J. Burczak and R. Granero-Belinchón,
On a generalized doubly parabolic keller–segel system in one spatial dimension, Mathematical Models and Methods in Applied Sciences, 26 (2016), 111-160.
doi: 10.1142/S0218202516500044. |
[5] |
P. Constantin, D. Córdoba, F. Gancedo, L. Rodriguez-Piazza and R. M. Strain,
On the muskat problem: Global in time results in 2d and 3d, American Journal of Mathematics, 138 (2016), 1455-1494.
doi: 10.1353/ajm.2016.0044. |
[6] |
D. Córdoba and F. Gancedo,
Contour dynamics of incompressible 3-d fluids in a porous medium with different densities, Communications in Mathematical Physics, 273 (2007), 445-471.
doi: 10.1007/s00220-007-0246-y. |
[7] |
F. Gancedo, E. Garcia-Juarez, N. Patel and R. M. Strain, On the muskat problem with viscosity jump: Global in time results, arXiv preprint, 2017, arXiv: 1710.11604. |
[8] |
Y. Gao, H. Ji, J.-G. Liu and T. P. Witelski,
A vicinal surface model for epitaxial growth with logarithmic free energy, AIMS, 23 (2018), 4433-4453.
doi: 10.3934/dcdsb.2018170. |
[9] |
Y. Gao, J.-G. Liu and J. Lu,
Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime, SIAM Journal on Mathematical Analysis, 49 (2017), 1705-1731.
doi: 10.1137/16M1094543. |
[10] |
Y. Gao, J.-G. Liu and X. Y. Lu, Gradient Flow Approach to an Exponential Thin Film Equation: Global Existence and Latent Singularity, ESAIM: COCV, Forthcoming article, 2018, arXiv: 1710.06995.
doi: 10.1051/cocv/2018037. |
[11] |
Y. Giga and R. V. Kohn,
Scale-invariant extinction time estimates for some singular diffusion equations, Discrete Contin. Dyn. Syst., 30 (2011), 509-535.
doi: 10.3934/dcds.2011.30.509. |
[12] |
J. Krug, H. T. Dobbs and S. Majaniemi, Adatom mobility for the solid-on-solid model, Zeitschrift für Physik B Condensed Matter, 97 (1995), 281–291.
doi: 10.1007/BF01307478. |
[13] |
J.-G. Liu and X. Xu,
Existence theorems for a multidimensional crystal surface model, SIAM Journal on Mathematical Analysis, 48 (2016), 3667-3687.
doi: 10.1137/16M1059400. |
[14] |
J. G. Liu and X. Xu,
Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions, SIAM Journal on Mathematical Analysis, 49 (2017), 2220-2245.
doi: 10.1137/16M1098474. |
[15] |
Jian-Guo Liu and Robert M. Strain., Global stability for solutions to the exponential PDE describing epitaxial growth., arXiv preprint arXiv: 1805.02246, 2018. |
[16] |
J. L. Marzuola and J. Weare, Relaxation of a family of broken-bond crystal-surface models, Physical Review E, 88 (2013), 032403.
doi: 10.1103/PhysRevE.88.032403. |
[17] |
N. Patel and R. M. Strain,
Large time decay estimates for the muskat equation, Communications in Partial Differential Equations, 42 (2017), 977-999.
doi: 10.1080/03605302.2017.1321661. |
[18] |
H. A. H. Shehadeh, R. V. Kohn and J. Weare,
The evolution of a crystal surface: Analysis of a one-dimensional step train connecting two facets in the adl regime, Physica D: Nonlinear Phenomena, 240 (2011), 1771-1784.
doi: 10.1016/j.physd.2011.07.016. |
[19] |
J. Simon,
Compact sets in the space $L^{p}(O, T; B)$, Annali di Matematica Pura ed Applicata, 146 (1986), 65-96.
doi: 10.1007/BF01762360. |
[20] |
X. Xu, Existence theorems for a crystal surface model involving the p-laplacian operator, arXiv preprint, 2017, arXiv: 1711.07405. |
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