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Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem
1. | Instituto de Matemática e Computação, Universidade Federal de Itajubá, MG, Brazil |
2. | Departamento de Matemática - DM, Universidade Federal de São Carlos, SP, Brazil |
In this article we consider a singularly perturbed Allen-Cahn problem $ u_t = \epsilon^2(a^2u_x)_x+b^2(u-u^3) $, for $ (x,t)\in (0,1)\times\mathbb{R}^+ $, supplied with no-flux boundary condition. The novelty here lies in the fact that the nonnegative spatial inhomogeneities $ a(\cdot) $ and $ b(\cdot) $ are allowed to vanish at some points in $ (0,1) $. Using the variational concept of $ \Gamma $-convergence we prove that, for $ \epsilon $ small, such degeneracy of $ a(\cdot) $ and $ b(\cdot) $ induces the existence of stable stationary solutions which develop internal transition layer as $ \epsilon\to 0 $.
References:
[1] |
S. Ai, X. Chen and S. P. Hastings,
Layers and spikes in non-homogeneous bistable reaction-diffusion equations, Transactions of the American Mathematical Society, 358 (2006), 3169-3206.
doi: 10.1090/S0002-9947-06-03834-7. |
[2] |
F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli,
Carleman estimates for degenerate parabolic operators with applications to null controllability, Journal of Evolution Equations, 6 (2006), 161-204.
doi: 10.1007/s00028-006-0222-6. |
[3] |
S. M. Allen and J. W. Cahn,
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[4] |
S. B. Angenent, J. Mallet-Paret and L. A. Peletier,
Stable transition layers in a semilinear boundary value problem, Journal of Differential Equations, 67 (1987), 212-242.
doi: 10.1016/0022-0396(87)90147-1. |
[5] |
F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964.
![]() ![]() |
[6] |
I. Boutaayamou, G. Fragnelli and L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions, Journal d'Analyse Mathématique, 135 (2018), 1–35.
doi: 10.1007/s11854-018-0030-2. |
[7] |
P. Caldiroli and R. Musina,
On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations and Applications, 7 (2000), 187-199.
doi: 10.1007/s000300050004. |
[8] |
M. Chipot and J. K. Hale, Stable equilibria with variable diffusion, Contemp. Math., 17, Amer. Math. Soc., Providence, RI, 1983, 209–213.
doi: 10.1090/conm/017/706100. |
[9] |
G. Dal Maso, An Introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, 8, 1993.
doi: 10.1007/978-1-4612-0327-8. |
[10] |
A. S. do Nascimento,
Local minimizers induced by spatial inhomogeneity with inner transition layer, Jounal of Differential Equations, 133 (1997), 203-223.
doi: 10.1006/jdeq.1996.3206. |
[11] |
L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, CRC Press, 2011.
doi: 10.1201/b10802.![]() ![]() ![]() |
[12] |
G. Fragnelli, G. R. Goldstein, J. A. Goldstein, and S. Romanelli, Generators with interior degeneracy on spaces of $L^2$-type, Electron, J. Differential Equations, (2012), No. 189, 30 pp. |
[13] |
G. Fusco and J. K. Hale,
Stable equilibria in a scalar parabolic equation with variable diffusion, SIAM Journal on Mathematical Analysis, 16 (1985), 1152-1164.
doi: 10.1137/0516085. |
[14] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, 80, Birkhäuser, Verlag, Basel, 1984.
doi: 10.1007/978-1-4684-9486-0. |
[15] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[16] |
N. I. Karachalios and N. B. Zographopoulos,
On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. Partial Differential Equations, 25 (2006), 361-393.
doi: 10.1007/s00526-005-0347-4. |
[17] |
R. V. Kohn and P. Sternberg,
Local minimizers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 69-84.
doi: 10.1017/S0308210500025026. |
[18] |
K. Kurata and H. Matsuzawa,
Multiple stable patterns in a balanced bistable equation with heterogeneous environments, Applicable Analysis, 89 (2010), 1023-1035.
doi: 10.1080/00036811003717947. |
[19] |
K. Nakashima,
Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Differential Equations, 191 (2003), 234-276.
doi: 10.1016/S0022-0396(02)00181-X. |
[20] |
K. Nakashima and K. Tanaka,
Clustering layers and boundary layers in spatially inhomogeneous phase transition problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 107-143.
doi: 10.1016/S0294-1449(02)00008-2. |
[21] |
J. Norbury and L.-C. Yeh,
The location and stability of interface solutions of an inhomogeneous parabolic problem, SIAM J. Appl. Math., 61 (2000/01), 1418-1430.
doi: 10.1137/S0036139997326491. |
[22] |
M. Sônego,
On the weakly degenerate Allen-Cahn equation, Advances in Nonlinear Analysis, 9 (2020), 361-371.
doi: 10.1515/anona-2020-0004. |
[23] |
P. Sternberg,
The effect of a singular perturbation on nonconvex variational problems, Archive for Rational Mechanics and Analysis, 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
[24] |
E. Yanagida,
Stability of stationary distributions in a space-dependent population growth process, J. Math. Biology, 15 (1982), 37-50.
doi: 10.1007/BF00275787. |
show all references
References:
[1] |
S. Ai, X. Chen and S. P. Hastings,
Layers and spikes in non-homogeneous bistable reaction-diffusion equations, Transactions of the American Mathematical Society, 358 (2006), 3169-3206.
doi: 10.1090/S0002-9947-06-03834-7. |
[2] |
F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli,
Carleman estimates for degenerate parabolic operators with applications to null controllability, Journal of Evolution Equations, 6 (2006), 161-204.
doi: 10.1007/s00028-006-0222-6. |
[3] |
S. M. Allen and J. W. Cahn,
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[4] |
S. B. Angenent, J. Mallet-Paret and L. A. Peletier,
Stable transition layers in a semilinear boundary value problem, Journal of Differential Equations, 67 (1987), 212-242.
doi: 10.1016/0022-0396(87)90147-1. |
[5] |
F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964.
![]() ![]() |
[6] |
I. Boutaayamou, G. Fragnelli and L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions, Journal d'Analyse Mathématique, 135 (2018), 1–35.
doi: 10.1007/s11854-018-0030-2. |
[7] |
P. Caldiroli and R. Musina,
On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations and Applications, 7 (2000), 187-199.
doi: 10.1007/s000300050004. |
[8] |
M. Chipot and J. K. Hale, Stable equilibria with variable diffusion, Contemp. Math., 17, Amer. Math. Soc., Providence, RI, 1983, 209–213.
doi: 10.1090/conm/017/706100. |
[9] |
G. Dal Maso, An Introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, 8, 1993.
doi: 10.1007/978-1-4612-0327-8. |
[10] |
A. S. do Nascimento,
Local minimizers induced by spatial inhomogeneity with inner transition layer, Jounal of Differential Equations, 133 (1997), 203-223.
doi: 10.1006/jdeq.1996.3206. |
[11] |
L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, CRC Press, 2011.
doi: 10.1201/b10802.![]() ![]() ![]() |
[12] |
G. Fragnelli, G. R. Goldstein, J. A. Goldstein, and S. Romanelli, Generators with interior degeneracy on spaces of $L^2$-type, Electron, J. Differential Equations, (2012), No. 189, 30 pp. |
[13] |
G. Fusco and J. K. Hale,
Stable equilibria in a scalar parabolic equation with variable diffusion, SIAM Journal on Mathematical Analysis, 16 (1985), 1152-1164.
doi: 10.1137/0516085. |
[14] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, 80, Birkhäuser, Verlag, Basel, 1984.
doi: 10.1007/978-1-4684-9486-0. |
[15] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[16] |
N. I. Karachalios and N. B. Zographopoulos,
On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. Partial Differential Equations, 25 (2006), 361-393.
doi: 10.1007/s00526-005-0347-4. |
[17] |
R. V. Kohn and P. Sternberg,
Local minimizers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 69-84.
doi: 10.1017/S0308210500025026. |
[18] |
K. Kurata and H. Matsuzawa,
Multiple stable patterns in a balanced bistable equation with heterogeneous environments, Applicable Analysis, 89 (2010), 1023-1035.
doi: 10.1080/00036811003717947. |
[19] |
K. Nakashima,
Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation, J. Differential Equations, 191 (2003), 234-276.
doi: 10.1016/S0022-0396(02)00181-X. |
[20] |
K. Nakashima and K. Tanaka,
Clustering layers and boundary layers in spatially inhomogeneous phase transition problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 107-143.
doi: 10.1016/S0294-1449(02)00008-2. |
[21] |
J. Norbury and L.-C. Yeh,
The location and stability of interface solutions of an inhomogeneous parabolic problem, SIAM J. Appl. Math., 61 (2000/01), 1418-1430.
doi: 10.1137/S0036139997326491. |
[22] |
M. Sônego,
On the weakly degenerate Allen-Cahn equation, Advances in Nonlinear Analysis, 9 (2020), 361-371.
doi: 10.1515/anona-2020-0004. |
[23] |
P. Sternberg,
The effect of a singular perturbation on nonconvex variational problems, Archive for Rational Mechanics and Analysis, 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
[24] |
E. Yanagida,
Stability of stationary distributions in a space-dependent population growth process, J. Math. Biology, 15 (1982), 37-50.
doi: 10.1007/BF00275787. |
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