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October  2021, 26(10): 5627-5640. doi: 10.3934/dcdsb.2020370

Stable transition layers in an unbalanced bistable equation

Maicon Sônego

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.

Keywords: Internal transition layers, unbalanced case, stability, semi-linear problem.
Mathematics Subject Classification: Primary: 35B25, 35B35; Secondary: 35B36, 35B40.
Citation: Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

E. De Giorgi, Convergence problems for functionals and operators, Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, (1979), 131–188.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, CRC press, 2011.  doi: 10.1201/b10802.  Google Scholar
[11]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 840, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

K. Nakashima, Stable transition layers in a balanced bistable equation, Differential and Integral Equations, 13 (2000), 1025–1038.  Google Scholar

[17]

M. Sônego, A note on interface formation in singularly perturbed elliptic problems. doi: 10.1080/17476933.2020.1825395.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

E. De Giorgi, Convergence problems for functionals and operators, Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, (1979), 131–188.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, CRC press, 2011.  doi: 10.1201/b10802.  Google Scholar
[11]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 840, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

K. Nakashima, Stable transition layers in a balanced bistable equation, Differential and Integral Equations, 13 (2000), 1025–1038.  Google Scholar

[17]

M. Sônego, A note on interface formation in singularly perturbed elliptic problems. doi: 10.1080/17476933.2020.1825395.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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