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  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 & Continuous Dynamical Systems - B, 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] $)
[1]

Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021006

[2]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[3]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014

[4]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035

[5]

Vakhtang Putkaradze, Stuart Rogers. Numerical simulations of a rolling ball robot actuated by internal point masses. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 143-207. doi: 10.3934/naco.2020021

[6]

Alexey Yulin, Alan Champneys. Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1341-1357. doi: 10.3934/dcdss.2011.4.1341

[7]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[8]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[9]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[10]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005

[11]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[12]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[13]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[14]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

[15]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045

[16]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[17]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[18]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[19]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[20]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (23)
  • HTML views (100)
  • Cited by (0)

Other articles
by authors

[Back to Top]