July  2021, 41(7): 3211-3240. doi: 10.3934/dcds.2020403

Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms

1. 

Departamento de Matemáticas y Mecánica, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Circuito Escolar s/n, Ciudad Universitaria C.P. 04510 Cd. Mx. (Mexico)

2. 

Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca' Foscari Venezia Mestre, Campus Scientifico, Via Torino 155, 30170 Venezia Mestre (Italy)

* Corresponding author

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

This paper establishes the emergence of slowly moving transition layer solutions for the
$ p $
-Laplacian (nonlinear) evolution equation,
$ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0, $
where
$ \varepsilon>0 $
and
$ p>1 $
are constants, driven by the action of a family of double-well potentials of the form
$ F(u) = \frac{1}{2\theta}|1-u^2|^\theta, $
indexed by
$ \theta>1 $
,
$ \theta\in \mathbb{R} $
with minima at two pure phases
$ u = \pm1 $
. The equation is endowed with initial conditions and boundary conditions of Neumann type. It is shown that interface layers, or solutions which initially are equal to
$ \pm 1 $
except at a finite number of thin transitions of width
$ \varepsilon $
, persist for an exponentially long time in the critical case with
$ \theta = p $
, and for an algebraically long time in the supercritical (or degenerate) case with
$ \theta>p $
. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg–Landau type are established. In contrast, in the subcritical case with
$ \theta<p $
, the transition layer solutions are stationary.
Citation: Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3211-3240. doi: 10.3934/dcds.2020403
References:
[1]

M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

J. BenediktP. GirgL. Kotrla and P. Takáč, Origin of the $p$-Laplacian and A. Missbach, Electron. J. Differ. Eq., 2018 (2018), 1-17.   Google Scholar

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F. Bethuel and D. Smets, Slow motion for equal depth multiple-well gradient systems: The degenerate case, Discrete Contin. Dyn. Syst., 33 (2013), 67-87.  doi: 10.3934/dcds.2013.33.67.  Google Scholar

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L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237.  doi: 10.1016/0022-0396(91)90147-2.  Google Scholar

[6]

J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t = \epsilon^2u_xx-f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576.  doi: 10.1002/cpa.3160420502.  Google Scholar

[7]

J. Carr and R. L. Pego, Invariant manifolds for metastable patterns in $u_t = \epsilon^2u_xx-f(u)$, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 133-160.  doi: 10.1017/S0308210500031425.  Google Scholar

[8]

M. S. ChangS. C. Lee and C.-C. Yen, Minimizers and gamma-convergence of energy functionals derived from $p$-Laplacian equation, Taiwanese J. Math., 13 (2009), 2021-2036.  doi: 10.11650/twjm/1500405655.  Google Scholar

[9]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[10]

X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differential Equations, 206 (2004), 399-437.  doi: 10.1016/j.jde.2004.05.017.  Google Scholar

[11]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.  doi: 10.1090/S0002-9947-1995-1672406-7.  Google Scholar

[12]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[13]

S. DipierroA. Farina and E. Valdinoci, Density estimates for degenerate double-well potentials, SIAM J. Math. Anal., 50 (2018), 6333-6347.  doi: 10.1137/17M114933X.  Google Scholar

[14]

S. DipierroA. Pinamonti and E. Valdinoci, Rigidity results for elliptic boundary value problems: Stable solutions for quasilinear equations with Neumann or Robin boundary conditions, Int. Math. Res. Not. IMRN, 2020 (2020), 1366-1384.  doi: 10.1093/imrn/rny055.  Google Scholar

[15]

L. EvansH. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.  doi: 10.1002/cpa.3160450903.  Google Scholar

[16]

R. Folino, Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension, J. Hyperbolic Differ. Equ., 14 (2017), 1-26.  doi: 10.1142/S0219891617500011.  Google Scholar

[17]

R. Folino, Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems, Electron. J. Differ. Eq., 2019 (2019), 1-21.   Google Scholar

[18]

R. Folino, C. A. Hernández Melo, L. F. López Ríos and R. G. Plaza, Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity,, Z. Angew. Math. Phys., 71 (2020), Paper No. 132, 25 pp. doi: 10.1007/s00033-020-01362-0.  Google Scholar

[19]

R. FolinoC. Lattanzio and C. Mascia, Slow dynamics for the hyperbolic Cahn-Hilliard equation in one-space dimension, Math. Methods Appl. Sci., 42 (2019), 2492-2512.  doi: 10.1002/mma.5525.  Google Scholar

[20]

R. Folino, C. Lattanzio, C. Mascia and M. Strani, Metastability for nonlinear convection-diffusion equations,, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 35, 20 pp. doi: 10.1007/s00030-017-0459-5.  Google Scholar

[21]

R. Folino, R. G. Plaza and M. Strani, Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion,, J. Math. Anal. Appl., 493 (2021), 124455, 29pp. doi: 10.1016/j.jmaa.2020.124455.  Google Scholar

[22]

G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations, 1 (1989), 75-94.  doi: 10.1007/BF01048791.  Google Scholar

[23]

C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34.  doi: 10.1137/S0036141092226053.  Google Scholar

[24]

E. J. Hurtado and M. Sônego, On the energy functionals derived from a non-homogeneous $p$-Laplacian equation: $\Gamma$-convergence, local minimizers and stable transition layers,, J. Math. Anal. Appl., 483 (2020), 123634, 13pp. doi: 10.1016/j.jmaa.2019.123634.  Google Scholar

[25]

J. G. L. Laforgue and R. E. O'Malley Jr., Shock layer movement for Burgers' equation, SIAM J. Appl. Math., 55 (1995), 332-347.  doi: 10.1137/S003613999326928X.  Google Scholar

[26]

K. A. LeeA. Petrosyan and J. L. Vázquez, Large-time geometric properties of solutions of the evolution $p$-Laplacian equation, J. Differential Equations, 229 (2006), 389-411.  doi: 10.1016/j.jde.2005.07.028.  Google Scholar

[27]

P. Lindqvist, Notes on the Stationary $p$-Laplace Equation, SpringerBriefs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-14501-9.  Google Scholar

[28]

J. L. Lions, Quelques Méthodes de Résolution des Problèemes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[29]

B. Lou, Singular limit of a $p$-Laplacian reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 110 (2003), 377-383.  doi: 10.1023/A:1021083015108.  Google Scholar

[30]

C. Mascia and M. Strani, Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws, SIAM J. Math. Anal., 45 (2013), 3084-3113.  doi: 10.1137/120875119.  Google Scholar

[31]

F. Otto and M. G. Reznikoff, Slow motion of gradient flows, J. Differential Equations, 237 (2007), 372-420.  doi: 10.1016/j.jde.2007.03.007.  Google Scholar

[32]

A. Petrosyan and E. Valdinoci, Density estimates for a degenerate/singular phase-transition model, SIAM J. Math. Anal., 36 (2005), 1057-1079.  doi: 10.1137/S0036141003437678.  Google Scholar

[33]

L. G. Reyna and M. J. Ward, On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math., 48 (1995), 79-120.  doi: 10.1002/cpa.3160480202.  Google Scholar

[34]

M. Strani, On the metastable behavior of solutions to a class of parabolic systems, Asymptot. Anal., 90 (2014), 325-344.  doi: 10.3233/ASY-141250.  Google Scholar

[35]

S. Takeuchi and Y. Yamada, Asymptotic properties of a reaction-diffusion equation with degenerate $p$-Laplacian, Nonlinear Anal., 42 (2000), 41-61.  doi: 10.1016/S0362-546X(98)00329-0.  Google Scholar

show all references

References:
[1]

M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[2]

J. BenediktP. GirgL. Kotrla and P. Takáč, Origin of the $p$-Laplacian and A. Missbach, Electron. J. Differ. Eq., 2018 (2018), 1-17.   Google Scholar

[3]

F. Bethuel and D. Smets, Slow motion for equal depth multiple-well gradient systems: The degenerate case, Discrete Contin. Dyn. Syst., 33 (2013), 67-87.  doi: 10.3934/dcds.2013.33.67.  Google Scholar

[4]

L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997.  doi: 10.1002/cpa.3160430804.  Google Scholar

[5]

L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237.  doi: 10.1016/0022-0396(91)90147-2.  Google Scholar

[6]

J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t = \epsilon^2u_xx-f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576.  doi: 10.1002/cpa.3160420502.  Google Scholar

[7]

J. Carr and R. L. Pego, Invariant manifolds for metastable patterns in $u_t = \epsilon^2u_xx-f(u)$, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 133-160.  doi: 10.1017/S0308210500031425.  Google Scholar

[8]

M. S. ChangS. C. Lee and C.-C. Yen, Minimizers and gamma-convergence of energy functionals derived from $p$-Laplacian equation, Taiwanese J. Math., 13 (2009), 2021-2036.  doi: 10.11650/twjm/1500405655.  Google Scholar

[9]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[10]

X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differential Equations, 206 (2004), 399-437.  doi: 10.1016/j.jde.2004.05.017.  Google Scholar

[11]

P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589.  doi: 10.1090/S0002-9947-1995-1672406-7.  Google Scholar

[12]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[13]

S. DipierroA. Farina and E. Valdinoci, Density estimates for degenerate double-well potentials, SIAM J. Math. Anal., 50 (2018), 6333-6347.  doi: 10.1137/17M114933X.  Google Scholar

[14]

S. DipierroA. Pinamonti and E. Valdinoci, Rigidity results for elliptic boundary value problems: Stable solutions for quasilinear equations with Neumann or Robin boundary conditions, Int. Math. Res. Not. IMRN, 2020 (2020), 1366-1384.  doi: 10.1093/imrn/rny055.  Google Scholar

[15]

L. EvansH. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.  doi: 10.1002/cpa.3160450903.  Google Scholar

[16]

R. Folino, Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension, J. Hyperbolic Differ. Equ., 14 (2017), 1-26.  doi: 10.1142/S0219891617500011.  Google Scholar

[17]

R. Folino, Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems, Electron. J. Differ. Eq., 2019 (2019), 1-21.   Google Scholar

[18]

R. Folino, C. A. Hernández Melo, L. F. López Ríos and R. G. Plaza, Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity,, Z. Angew. Math. Phys., 71 (2020), Paper No. 132, 25 pp. doi: 10.1007/s00033-020-01362-0.  Google Scholar

[19]

R. FolinoC. Lattanzio and C. Mascia, Slow dynamics for the hyperbolic Cahn-Hilliard equation in one-space dimension, Math. Methods Appl. Sci., 42 (2019), 2492-2512.  doi: 10.1002/mma.5525.  Google Scholar

[20]

R. Folino, C. Lattanzio, C. Mascia and M. Strani, Metastability for nonlinear convection-diffusion equations,, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 35, 20 pp. doi: 10.1007/s00030-017-0459-5.  Google Scholar

[21]

R. Folino, R. G. Plaza and M. Strani, Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion,, J. Math. Anal. Appl., 493 (2021), 124455, 29pp. doi: 10.1016/j.jmaa.2020.124455.  Google Scholar

[22]

G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations, 1 (1989), 75-94.  doi: 10.1007/BF01048791.  Google Scholar

[23]

C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34.  doi: 10.1137/S0036141092226053.  Google Scholar

[24]

E. J. Hurtado and M. Sônego, On the energy functionals derived from a non-homogeneous $p$-Laplacian equation: $\Gamma$-convergence, local minimizers and stable transition layers,, J. Math. Anal. Appl., 483 (2020), 123634, 13pp. doi: 10.1016/j.jmaa.2019.123634.  Google Scholar

[25]

J. G. L. Laforgue and R. E. O'Malley Jr., Shock layer movement for Burgers' equation, SIAM J. Appl. Math., 55 (1995), 332-347.  doi: 10.1137/S003613999326928X.  Google Scholar

[26]

K. A. LeeA. Petrosyan and J. L. Vázquez, Large-time geometric properties of solutions of the evolution $p$-Laplacian equation, J. Differential Equations, 229 (2006), 389-411.  doi: 10.1016/j.jde.2005.07.028.  Google Scholar

[27]

P. Lindqvist, Notes on the Stationary $p$-Laplace Equation, SpringerBriefs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-14501-9.  Google Scholar

[28]

J. L. Lions, Quelques Méthodes de Résolution des Problèemes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[29]

B. Lou, Singular limit of a $p$-Laplacian reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 110 (2003), 377-383.  doi: 10.1023/A:1021083015108.  Google Scholar

[30]

C. Mascia and M. Strani, Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws, SIAM J. Math. Anal., 45 (2013), 3084-3113.  doi: 10.1137/120875119.  Google Scholar

[31]

F. Otto and M. G. Reznikoff, Slow motion of gradient flows, J. Differential Equations, 237 (2007), 372-420.  doi: 10.1016/j.jde.2007.03.007.  Google Scholar

[32]

A. Petrosyan and E. Valdinoci, Density estimates for a degenerate/singular phase-transition model, SIAM J. Math. Anal., 36 (2005), 1057-1079.  doi: 10.1137/S0036141003437678.  Google Scholar

[33]

L. G. Reyna and M. J. Ward, On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math., 48 (1995), 79-120.  doi: 10.1002/cpa.3160480202.  Google Scholar

[34]

M. Strani, On the metastable behavior of solutions to a class of parabolic systems, Asymptot. Anal., 90 (2014), 325-344.  doi: 10.3233/ASY-141250.  Google Scholar

[35]

S. Takeuchi and Y. Yamada, Asymptotic properties of a reaction-diffusion equation with degenerate $p$-Laplacian, Nonlinear Anal., 42 (2000), 41-61.  doi: 10.1016/S0362-546X(98)00329-0.  Google Scholar

Figure 1.  Plots of the potential function (1.4) for $ \theta = 2,4,6 $, which underly different behaviors when compared to the diffusion parameter $ p\geq2 $. For example, when $ p = 4 $ the former cases correspond to subcritical ($ \theta<p $), critical ($ \theta = p $) and supercritical or degenerate ($ \theta>p $) cases, respectively
Figure 2.  The level of the energy (given by the constant $ C $) has to be such that $ C> -F(u) $. In particular, bounded nontrivial solutions can be found only for $ C\in(-1/2\theta, 0) $. When $ C = 0 $ we have the heteroclinic solution (that touches the value $ \pm 1 $ only in the case $ \theta<p $)
Figure 3.  Solutions to (1.3) for $ \varepsilon = 0.1 $, $ p = \theta = 2 $ (left), and $ p = \theta = 4 $ (right); the initial datum $ u_0 $ is as in (3.36) with $ 6 $ transition points located at $ (-3.4,-2,-0.5, 0.8, 2.2, 3.2) $
Figure 3, while the potential $ F $ is as in (4.1) with $ \theta = 4 $">Figure 4.  Solutions to (1.3) for $ \varepsilon = 0.1 $, $ p = 2 $ (left) and $ p = 3 $ (right); the initial datum $ u_0 $ is as in Figure 3, while the potential $ F $ is as in (4.1) with $ \theta = 4 $
Figure 3">Figure 5.  Solutions to (1.3) for $ \varepsilon = 0.1 $, $ \theta = 8 $ and $ p = \pi $ (left) and $ p = 5.5 $ (right); the initial datum $ u_0 $ is as in Figure 3
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