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June  2022, 27(6): 3297-3311. doi: 10.3934/dcdsb.2021185

Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem

Maicon Sônego 1, and  Arnaldo Simal do Nascimento 2,

1. 

Instituto de Matemática e Computação, Universidade Federal de Itajubá, MG, Brazil
2. 

Departamento de Matemática - DM, Universidade Federal de São Carlos, SP, Brazil

Received  January 2021 Revised  May 2021 Published  June 2022 Early access  July 2021

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 $.

Keywords: Stable transition layer, degenerate problem, $ \Gamma $-convergence, Allen-Cahn equation.
Mathematics Subject Classification: Primary: 35K65; Secondary: 35K57, 35B25, 35B35.
Citation: Maicon Sônego, Arnaldo Simal do Nascimento. Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3297-3311. doi: 10.3934/dcdsb.2021185
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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