doi: 10.3934/dcdsb.2021226
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Boundary layer solutions to singularly perturbed quasilinear systems

1. 

Department of Mathematics, Faculty of Physics, Moscow State University, Vorob'jovy Gory, 19899 Moscow, Russia

2. 

Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam, Germany

3. 

Insitute of Mathematics, Humboldt University of Berlin, Rudower Chaussee 25, 12489 Berlin, Germany

* Corresponding author

Received  November 2020 Revised  August 2021 Early access September 2021

We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type $ \varepsilon^2\left(A(x, u(x), \varepsilon)u'(x)\right)' = f(x, u(x), \varepsilon) $. The new features are that we do not consider one scalar equation, but systems, that the systems are allowed to be quasilinear, and that the systems are spatially non-smooth. Although the results about existence, asymptotic behavior, local uniqueness and stability of boundary layer solutions are similar to those known for semilinear, scalar and smooth problems, there are at least three essential differences. First, the asymptotic convergence rates valid for smooth problems are not true anymore, in general, in the non-smooth case. Second, a specific local uniqueness condition from the scalar case is not sufficient anymore in the vectorial case. And third, the monotonicity condition, which is sufficient for stability of boundary layers in the scalar case, must be adjusted to the vectorial case.

Citation: Valentin Butuzov, Nikolay Nefedov, Oleh Omel'chenko, Lutz Recke. Boundary layer solutions to singularly perturbed quasilinear systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021226
References:
[1]

P. W. Bates and J. Shi, Existence and stability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264.  doi: 10.1016/S0022-1236(02)00013-7.  Google Scholar

[2]

M. S. Berger and L. E. Fraenkel, On singular perturbations of nonlinear operator equations, Indiana Univ. Math. J., 20 (1970/71), 623-631.  doi: 10.1512/iumj.1971.20.20050.  Google Scholar

[3]

V. F. Butuzov, Asymptotics of the solution of a system of singularly perturbed equations in the case of a multiple root of the degenerate equation, Differ. Equations, 50 (2014), 177-188.  doi: 10.1134/S0012266114020050.  Google Scholar

[4]

V. F. ButuzovN. N. NefedovL. Recke and K. R. Schneider, On a singularly perturbed initial value problem in the case of a double root of the degenerate equation, Nonlinear Anal., Theory, Methods, Appl., Ser. A, 83 (2013), 1-11.  doi: 10.1016/j.na.2013.01.013.  Google Scholar

[5]

V. F. Butuzov, N. N. Nefedov, O. E. Omel'chenko, L. Recke and K. R. Schneider, An implicit function theorem and applications to nonsmooth boundary layers, in Patterns of Dynamics (eds. P. Gurevich et al.), Springer Proc. in Mathematics & Statistics vol. 205, (2017), 111–127. doi: 10.1007/978-3-319-64173-7_7.  Google Scholar

[6]

M. ChandruP. Das and H. Ramos, Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data, Math. Methods Appl. Sci., 41 (2018), 5359-5387.  doi: 10.1002/mma.5067.  Google Scholar

[7]

Ju. L. , Stability of Solutions of Differential Equations in Banach Spaces, AMS, Providence, 1974.  Google Scholar

[8]

M. del Pino and J. Wei, An introduction to the finite and infinite dimensional reduction methods, in Geometric Analysis Around Scalar Curvatures (eds. Fei Han et al.), World Scientific Lecture Notes Series 31, Institute for Mathematical Sciences, National University of Singapore, (2016), 35–118.  Google Scholar

[9]

P. C. Fife and W. M. Greenlee, Interior transition layers for elliptic boundary value problems with a small parameter, Russ. Math. Surv., 29 (1974), 103-130.  doi: 10.1070/RM1974v029n04ABEH001291.  Google Scholar

[10]

P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), 497-521.  doi: 10.1016/0022-247X(76)90218-3.  Google Scholar

[11]

G.-M. Gie, M. Hamouda, Ch.-Y. Jung and R. M. Temam, Singular Perturbations and Boundary Layers, Appl. Math. Sciences, vol. 200, Springer, 2018. doi: 10.1007/978-3-030-00638-9.  Google Scholar

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W. Hackbusch, Integral Equations. Theory and Numerical Treatment, ISNM Int. Series Num. Math., vol. 120, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9215-5.  Google Scholar

[13]

J. K. Hale and K. Sakamoto, A Lyapunov-Schmidt method for transition layers in reaction-diffusion systems, Hiroshima Math. J., 35 (2005), 205-249.   Google Scholar

[14]

J. K. Hale and D. Salazar, Boundary layers in a semilinear parabolic problem, Tohoku Math. J., 51 (1999), 421-432.  doi: 10.2748/tmj/1178224771.  Google Scholar

[15]

D. Henry, Geometric Theory of Semilinear parabolic Equations, Lecture Notes in Math., vol. 840, Springer, 1981.  Google Scholar

[16]

Y. LatushkinJ. Prüss and R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions, J. Evol. Equ., 6 (2006), 537-576.  doi: 10.1007/s00028-006-0272-9.  Google Scholar

[17]

N. T. LevashovaN. N. NefedovO. A. NikolaevaA. O. Orlov and A. A. Panin, The solution with internal transition layer of the reaction-diffusion equation in the case of discontinuous reactive and diffusive terms, Math. Methods Appl. Sci., 41 (2018), 9203-9217.  doi: 10.1002/mma.5134.  Google Scholar

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X.-B. Lin, Asymptotic expansion for layer solutions of a singularly perturbed reaction-diffusion system, Trans. AMS, 348 (1996), 713-753.  doi: 10.1090/S0002-9947-96-01542-5.  Google Scholar

[19]

X.-B. Lin, Construction and asymptotic stability of structurally stable internal layer solutions, Trans. AMS, 353 (2001), 2983-3043.  doi: 10.1090/S0002-9947-01-02769-6.  Google Scholar

[20]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonl. Differential Equations, vol. 246, Birkhäuser, 1995.  Google Scholar

[21]

R. Magnus, The implicit function theorem and multi-bump solutions of periodic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 559-583.  doi: 10.1017/S0308210500005060.  Google Scholar

[22]

B.-V. Matioc and Ch. Walker, On the principle of linearized stability in interpolation spaces for quasilinear evolution equations, Monatsh. Math., 191 (2020), 615-634.  doi: 10.1007/s00605-019-01352-z.  Google Scholar

[23]

M. NiY. PengN. T. Levashova and O. A. Nikolaeva, Internal layers for a singularly perturbed second-order quasilinear differential equation with discontinuous right-hand side, Differ. Equations, 53 (2017), 1567-1577.  doi: 10.1134/S0012266117120059.  Google Scholar

[24]

N. N. Nefedov and M. Ni, Internal layers in the one-dimensional reaction-diffusion equation with a discontinuous reactive term, Comput. Math. Math. Phys., 55 (2015), 2001-2007.  doi: 10.1134/S096554251512012X.  Google Scholar

[25]

O. Omel'chenko and L. Recke, Boundary layer solutions to singularly perturbed problems via the implicit function theorem, Asymptot. Anal., 62 (2009), 207-225.  doi: 10.3233/ASY-2009-0921.  Google Scholar

[26]

O. Omel'chenko and L. Recke, Existence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problems, Hiroshima Math. J., 45 (2015), 35-89.   Google Scholar

[27]

O. E. Omel'chenkoL. ReckeV. F. Butuzov and N. N. Nefedov, Time-periodic boundary layer solutions to singularly perturbed parabolic problems, J. Differ. Equations, 262 (2017), 4823-4862.  doi: 10.1016/j.jde.2016.12.020.  Google Scholar

[28]

E. O'Riordan, Interior layers in singularly perturbed problems, in Differential Equations and Numerical Analysis (eds. V. Sigamini et al.), Springer Proc. in Mathematics & Statistics vol. 172, (2016), 25–40. doi: 10.1007/978-81-322-3598-9_2.  Google Scholar

[29]

J. PrüssG. Simonett and R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems, J. Differ. Equations, 246 (2009), 3902-3931.  doi: 10.1016/j.jde.2008.10.034.  Google Scholar

[30]

L. Recke, Use of very weak approximate boundary layer solutions to spatially nonsmooth singularly perturbed problems, J. Math. Anal. Appl., 506 (2022), 125552. doi: 10.1016/j.jmaa.2021.125552.  Google Scholar

[31]

L. Recke and O. E. Omel'chenko, Boundary layer solutions to problems with infinite dimensional singular and regular perturbations, J. Differ. Equations, 245 (2008), 3806-3822.  doi: 10.1016/j.jde.2008.01.017.  Google Scholar

[32]

Ch. Sourdis, Analysis of an irregular boundary layer behaviour for the steady state flow of a Boussinesq fluid, Discrete Contin. Dyn. Syst., 37 (2017), 1039-1059.  doi: 10.3934/dcds.2017043.  Google Scholar

[33]

M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems, Adv. Differ. Equ., 8 (2003), 29-54.   Google Scholar

[34]

J. Wei and M. Winter, Stability of cluster solutions in a cooperative consumer chain model, J. Math. Biol., 68 (2014), 1-39.  doi: 10.1007/s00285-012-0616-8.  Google Scholar

show all references

References:
[1]

P. W. Bates and J. Shi, Existence and stability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264.  doi: 10.1016/S0022-1236(02)00013-7.  Google Scholar

[2]

M. S. Berger and L. E. Fraenkel, On singular perturbations of nonlinear operator equations, Indiana Univ. Math. J., 20 (1970/71), 623-631.  doi: 10.1512/iumj.1971.20.20050.  Google Scholar

[3]

V. F. Butuzov, Asymptotics of the solution of a system of singularly perturbed equations in the case of a multiple root of the degenerate equation, Differ. Equations, 50 (2014), 177-188.  doi: 10.1134/S0012266114020050.  Google Scholar

[4]

V. F. ButuzovN. N. NefedovL. Recke and K. R. Schneider, On a singularly perturbed initial value problem in the case of a double root of the degenerate equation, Nonlinear Anal., Theory, Methods, Appl., Ser. A, 83 (2013), 1-11.  doi: 10.1016/j.na.2013.01.013.  Google Scholar

[5]

V. F. Butuzov, N. N. Nefedov, O. E. Omel'chenko, L. Recke and K. R. Schneider, An implicit function theorem and applications to nonsmooth boundary layers, in Patterns of Dynamics (eds. P. Gurevich et al.), Springer Proc. in Mathematics & Statistics vol. 205, (2017), 111–127. doi: 10.1007/978-3-319-64173-7_7.  Google Scholar

[6]

M. ChandruP. Das and H. Ramos, Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data, Math. Methods Appl. Sci., 41 (2018), 5359-5387.  doi: 10.1002/mma.5067.  Google Scholar

[7]

Ju. L. , Stability of Solutions of Differential Equations in Banach Spaces, AMS, Providence, 1974.  Google Scholar

[8]

M. del Pino and J. Wei, An introduction to the finite and infinite dimensional reduction methods, in Geometric Analysis Around Scalar Curvatures (eds. Fei Han et al.), World Scientific Lecture Notes Series 31, Institute for Mathematical Sciences, National University of Singapore, (2016), 35–118.  Google Scholar

[9]

P. C. Fife and W. M. Greenlee, Interior transition layers for elliptic boundary value problems with a small parameter, Russ. Math. Surv., 29 (1974), 103-130.  doi: 10.1070/RM1974v029n04ABEH001291.  Google Scholar

[10]

P. C. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations, J. Math. Anal. Appl., 54 (1976), 497-521.  doi: 10.1016/0022-247X(76)90218-3.  Google Scholar

[11]

G.-M. Gie, M. Hamouda, Ch.-Y. Jung and R. M. Temam, Singular Perturbations and Boundary Layers, Appl. Math. Sciences, vol. 200, Springer, 2018. doi: 10.1007/978-3-030-00638-9.  Google Scholar

[12]

W. Hackbusch, Integral Equations. Theory and Numerical Treatment, ISNM Int. Series Num. Math., vol. 120, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9215-5.  Google Scholar

[13]

J. K. Hale and K. Sakamoto, A Lyapunov-Schmidt method for transition layers in reaction-diffusion systems, Hiroshima Math. J., 35 (2005), 205-249.   Google Scholar

[14]

J. K. Hale and D. Salazar, Boundary layers in a semilinear parabolic problem, Tohoku Math. J., 51 (1999), 421-432.  doi: 10.2748/tmj/1178224771.  Google Scholar

[15]

D. Henry, Geometric Theory of Semilinear parabolic Equations, Lecture Notes in Math., vol. 840, Springer, 1981.  Google Scholar

[16]

Y. LatushkinJ. Prüss and R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions, J. Evol. Equ., 6 (2006), 537-576.  doi: 10.1007/s00028-006-0272-9.  Google Scholar

[17]

N. T. LevashovaN. N. NefedovO. A. NikolaevaA. O. Orlov and A. A. Panin, The solution with internal transition layer of the reaction-diffusion equation in the case of discontinuous reactive and diffusive terms, Math. Methods Appl. Sci., 41 (2018), 9203-9217.  doi: 10.1002/mma.5134.  Google Scholar

[18]

X.-B. Lin, Asymptotic expansion for layer solutions of a singularly perturbed reaction-diffusion system, Trans. AMS, 348 (1996), 713-753.  doi: 10.1090/S0002-9947-96-01542-5.  Google Scholar

[19]

X.-B. Lin, Construction and asymptotic stability of structurally stable internal layer solutions, Trans. AMS, 353 (2001), 2983-3043.  doi: 10.1090/S0002-9947-01-02769-6.  Google Scholar

[20]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonl. Differential Equations, vol. 246, Birkhäuser, 1995.  Google Scholar

[21]

R. Magnus, The implicit function theorem and multi-bump solutions of periodic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 559-583.  doi: 10.1017/S0308210500005060.  Google Scholar

[22]

B.-V. Matioc and Ch. Walker, On the principle of linearized stability in interpolation spaces for quasilinear evolution equations, Monatsh. Math., 191 (2020), 615-634.  doi: 10.1007/s00605-019-01352-z.  Google Scholar

[23]

M. NiY. PengN. T. Levashova and O. A. Nikolaeva, Internal layers for a singularly perturbed second-order quasilinear differential equation with discontinuous right-hand side, Differ. Equations, 53 (2017), 1567-1577.  doi: 10.1134/S0012266117120059.  Google Scholar

[24]

N. N. Nefedov and M. Ni, Internal layers in the one-dimensional reaction-diffusion equation with a discontinuous reactive term, Comput. Math. Math. Phys., 55 (2015), 2001-2007.  doi: 10.1134/S096554251512012X.  Google Scholar

[25]

O. Omel'chenko and L. Recke, Boundary layer solutions to singularly perturbed problems via the implicit function theorem, Asymptot. Anal., 62 (2009), 207-225.  doi: 10.3233/ASY-2009-0921.  Google Scholar

[26]

O. Omel'chenko and L. Recke, Existence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problems, Hiroshima Math. J., 45 (2015), 35-89.   Google Scholar

[27]

O. E. Omel'chenkoL. ReckeV. F. Butuzov and N. N. Nefedov, Time-periodic boundary layer solutions to singularly perturbed parabolic problems, J. Differ. Equations, 262 (2017), 4823-4862.  doi: 10.1016/j.jde.2016.12.020.  Google Scholar

[28]

E. O'Riordan, Interior layers in singularly perturbed problems, in Differential Equations and Numerical Analysis (eds. V. Sigamini et al.), Springer Proc. in Mathematics & Statistics vol. 172, (2016), 25–40. doi: 10.1007/978-81-322-3598-9_2.  Google Scholar

[29]

J. PrüssG. Simonett and R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems, J. Differ. Equations, 246 (2009), 3902-3931.  doi: 10.1016/j.jde.2008.10.034.  Google Scholar

[30]

L. Recke, Use of very weak approximate boundary layer solutions to spatially nonsmooth singularly perturbed problems, J. Math. Anal. Appl., 506 (2022), 125552. doi: 10.1016/j.jmaa.2021.125552.  Google Scholar

[31]

L. Recke and O. E. Omel'chenko, Boundary layer solutions to problems with infinite dimensional singular and regular perturbations, J. Differ. Equations, 245 (2008), 3806-3822.  doi: 10.1016/j.jde.2008.01.017.  Google Scholar

[32]

Ch. Sourdis, Analysis of an irregular boundary layer behaviour for the steady state flow of a Boussinesq fluid, Discrete Contin. Dyn. Syst., 37 (2017), 1039-1059.  doi: 10.3934/dcds.2017043.  Google Scholar

[33]

M. Taniguchi, A uniform convergence theorem for singular limit eigenvalue problems, Adv. Differ. Equ., 8 (2003), 29-54.   Google Scholar

[34]

J. Wei and M. Winter, Stability of cluster solutions in a cooperative consumer chain model, J. Math. Biol., 68 (2014), 1-39.  doi: 10.1007/s00285-012-0616-8.  Google Scholar

Figure 1.  The homoclinic solution to problem (7.2)
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