Figure 1.
$ u_{\epsilon} $ developing two internal transition layer with interfaces at $ \overline{x}_1 $ and $ \overline{x}_2 $ (isolated local minima of $ \gamma $ in $ [x_1,x_2] $ )
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October
2021, 26(10): 5627-5640.
doi: 10.3934/dcdsb.2020370
Stable transition layers in an unbalanced bistable equation
Instituto de Matemática e Computação, Universidade Federal de Itajubá, MG, Brazil |
In this paper we are concerned with the existence of stable stationary solutions for the problem $ u_t = \epsilon^2(k_1^2(x) u_x)_x+k_2^2(x)g(u,x) $, $ (t,x)\in\mathbb{R}^+\times (0,1) $ subject to Neumann boundary condition. We suppose that $ k_1,k_2\in C^1(0,1) $ are positive functions and $ g $ is an unbalanced bistable function. We prove the existence of a family of stable stationary solutions developing internal transition layers in a specific sub-interval of $ (0,1) $. For this, we provide a general variational method inspired by the $ \Gamma $-convergence theory.
Mathematics Subject Classification:
Primary: 35B25, 35B35; Secondary: 35B36, 35B40.
Citation:
Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete and Continuous Dynamical Systems - B,
2021, 26
(10)
: 5627-5640.
doi: 10.3934/dcdsb.2020370
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References:
| [1] |
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. |
| [2] |
E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem, Journal of Differential Equations, 194 (2003), 382–405.
doi: 10.1016/S0022-0396(03)00176-1. |
| [3] |
E. N. Dancer and S. Yan, Construction of various types of solutions for an elliptic problem, Calculus of Variations and Partial Differential Equations, 20 (2004), 93–118.
doi: 10.1007/s00526-003-0229-6. |
| [4] |
E. De Giorgi, Convergence problems for functionals and operators, Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, (1979), 131–188. |
| [5] |
A. S. do Nascimento, Stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in N-dimensional domains, Journal of Differential Equations, 190 (2003) 16–38.
doi: 10.1016/S0022-0396(02)00147-X. |
| [6] |
A. S. do Nascimento, Inner transition layers in a elliptic boundary value problem: a necessary condition, Nonlinear Analysis: Theory, Methods and Applications, 44 (2001), 487–497.
doi: 10.1016/S0362-546X(99)00276-X. |
| [7] |
A. S. do Nascimento and M. Sônego, Stable Transition Layers to Singularly Perturbed Spatially Inhomogeneous Allen-Cahn Equation, Advanced Nonlinear Studies, 15 (2015), 363–376.
doi: 10.1515/ans-2015-0205. |
| [8] |
A. S. do Nascimento and R. J. de Moura, Layered stable equilibria of a reaction-diffusion equation with nonlinear Neumann boundary condition, J. Math. Anal. Appl., 347 (2008), 123–135.
doi: 10.1016/j.jmaa.2008.06.001. |
| [9] |
A. S. do Nascimento and M. Sônego, Stable equilibria of a singularly perturbed reaction-diffusion equation when the roots of the degenerate equation contact or intersect along a non-smooth hypersurface, Journal of Evolution Equations, 16 (2016), 317–339.
doi: 10.1007/s00028-015-0304-4. |
| [10] |
L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, CRC press, 2011.
doi: 10.1201/b10802.
|
| [11] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984.
doi: 10.1007/978-1-4684-9486-0. |
| [12] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 840, 1981. |
| [13] |
F. Mahmoudi, A. Malchiodi and J. Wei, Transition layer for the heterogeneous Allen-Cahn equation, Ann. I. H. Poincare. AN, 25 (2008), 609–631.
doi: 10.1016/j.anihpc.2007.03.008. |
| [14] |
H. Matsuzawa, Stable transition layers in a balanced bistable equation with degeneracy, Nonlinear Analysis, 58 (2004), 45–67.
doi: 10.1016/j.na.2004.04.006. |
| [15] |
H. Matsuzawa, Asymptotic profile of a radially symmetric solution with transition layers for an unbalanced bistable equation, Electronic Journal of Differential Equations, (2006), 1–12. |
| [16] |
K. Nakashima, Stable transition layers in a balanced bistable equation, Differential and Integral Equations, 13 (2000), 1025–1038. |
| [17] |
M. Sônego, A note on interface formation in singularly perturbed elliptic problems.
doi: 10.1080/17476933.2020.1825395. |
| [18] |
M. Sônego, Patterns in a balanced bistable equation with hehterogeneous environments on surfaces of revolution, Differ. Equ. Appl., 8 (2016), 521–533.
doi: 10.7153/dea-08-29. |
| [19] |
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. |
show all references
References:
| [1] |
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. |
| [2] |
E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem, Journal of Differential Equations, 194 (2003), 382–405.
doi: 10.1016/S0022-0396(03)00176-1. |
| [3] |
E. N. Dancer and S. Yan, Construction of various types of solutions for an elliptic problem, Calculus of Variations and Partial Differential Equations, 20 (2004), 93–118.
doi: 10.1007/s00526-003-0229-6. |
| [4] |
E. De Giorgi, Convergence problems for functionals and operators, Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, (1979), 131–188. |
| [5] |
A. S. do Nascimento, Stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in N-dimensional domains, Journal of Differential Equations, 190 (2003) 16–38.
doi: 10.1016/S0022-0396(02)00147-X. |
| [6] |
A. S. do Nascimento, Inner transition layers in a elliptic boundary value problem: a necessary condition, Nonlinear Analysis: Theory, Methods and Applications, 44 (2001), 487–497.
doi: 10.1016/S0362-546X(99)00276-X. |
| [7] |
A. S. do Nascimento and M. Sônego, Stable Transition Layers to Singularly Perturbed Spatially Inhomogeneous Allen-Cahn Equation, Advanced Nonlinear Studies, 15 (2015), 363–376.
doi: 10.1515/ans-2015-0205. |
| [8] |
A. S. do Nascimento and R. J. de Moura, Layered stable equilibria of a reaction-diffusion equation with nonlinear Neumann boundary condition, J. Math. Anal. Appl., 347 (2008), 123–135.
doi: 10.1016/j.jmaa.2008.06.001. |
| [9] |
A. S. do Nascimento and M. Sônego, Stable equilibria of a singularly perturbed reaction-diffusion equation when the roots of the degenerate equation contact or intersect along a non-smooth hypersurface, Journal of Evolution Equations, 16 (2016), 317–339.
doi: 10.1007/s00028-015-0304-4. |
| [10] |
L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, CRC press, 2011.
doi: 10.1201/b10802.
|
| [11] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984.
doi: 10.1007/978-1-4684-9486-0. |
| [12] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 840, 1981. |
| [13] |
F. Mahmoudi, A. Malchiodi and J. Wei, Transition layer for the heterogeneous Allen-Cahn equation, Ann. I. H. Poincare. AN, 25 (2008), 609–631.
doi: 10.1016/j.anihpc.2007.03.008. |
| [14] |
H. Matsuzawa, Stable transition layers in a balanced bistable equation with degeneracy, Nonlinear Analysis, 58 (2004), 45–67.
doi: 10.1016/j.na.2004.04.006. |
| [15] |
H. Matsuzawa, Asymptotic profile of a radially symmetric solution with transition layers for an unbalanced bistable equation, Electronic Journal of Differential Equations, (2006), 1–12. |
| [16] |
K. Nakashima, Stable transition layers in a balanced bistable equation, Differential and Integral Equations, 13 (2000), 1025–1038. |
| [17] |
M. Sônego, A note on interface formation in singularly perturbed elliptic problems.
doi: 10.1080/17476933.2020.1825395. |
| [18] |
M. Sônego, Patterns in a balanced bistable equation with hehterogeneous environments on surfaces of revolution, Differ. Equ. Appl., 8 (2016), 521–533.
doi: 10.7153/dea-08-29. |
| [19] |
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. |
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