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

Received  July 2020 Revised  October 2020 Published  October 2021 Early access  December 2020

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.

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
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.

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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