# American Institute of Mathematical Sciences

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
 $\theta , the transition layer solutions are stationary. Citation: Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to$ p $-Laplacian diffusion problems with bistable reaction terms. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3211-3240. doi: 10.3934/dcds.2020403 ##### References:  [1] M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2. [2] J. Benedikt, P. Girg, L. Kotrla and P. Takáč, Origin of the$p$-Laplacian and A. Missbach, Electron. J. Differ. Eq., 2018 (2018), 1-17. [3] F. Bethuel and D. Smets, Slow motion for equal depth multiple-well gradient systems: The degenerate case, Discrete Contin. Dyn. Syst., 33 (2013), 67-87. doi: 10.3934/dcds.2013.33.67. [4] L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997. doi: 10.1002/cpa.3160430804. [5] L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237. doi: 10.1016/0022-0396(91)90147-2. [6] J. Carr and R. L. Pego, Metastable patterns in solutions of$u_t = \epsilon^2u_xx-f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576. doi: 10.1002/cpa.3160420502. [7] J. Carr and R. L. Pego, Invariant manifolds for metastable patterns in$u_t = \epsilon^2u_xx-f(u)$, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 133-160. doi: 10.1017/S0308210500031425. [8] M. S. Chang, S. C. Lee and C.-C. Yen, Minimizers and gamma-convergence of energy functionals derived from$p$-Laplacian equation, Taiwanese J. Math., 13 (2009), 2021-2036. doi: 10.11650/twjm/1500405655. [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. [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. [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. [12] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [13] S. Dipierro, A. Farina and E. Valdinoci, Density estimates for degenerate double-well potentials, SIAM J. Math. Anal., 50 (2018), 6333-6347. doi: 10.1137/17M114933X. [14] S. Dipierro, A. 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. [15] L. Evans, H. 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. [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. [17] R. Folino, Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems, Electron. J. Differ. Eq., 2019 (2019), 1-21. [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. [19] R. Folino, C. 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. [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. [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. [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. [23] C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34. doi: 10.1137/S0036141092226053. [24] E. J. Hurtado and M. Sônego, On the energy functionals derived from a non-homogeneous$p$-Laplacian equation:$\Gamma$-convergence, local minimizers and stable transition layers,, J. Math. Anal. Appl., 483 (2020), 123634, 13pp. doi: 10.1016/j.jmaa.2019.123634. [25] J. G. L. Laforgue and R. E. O'Malley Jr., Shock layer movement for Burgers' equation, SIAM J. Appl. Math., 55 (1995), 332-347. doi: 10.1137/S003613999326928X. [26] K. A. Lee, A. Petrosyan and J. L. Vázquez, Large-time geometric properties of solutions of the evolution$p$-Laplacian equation, J. Differential Equations, 229 (2006), 389-411. doi: 10.1016/j.jde.2005.07.028. [27] P. Lindqvist, Notes on the Stationary$p$-Laplace Equation, SpringerBriefs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-14501-9. [28] J. L. Lions, Quelques Méthodes de Résolution des Problèemes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. [29] B. Lou, Singular limit of a$p$-Laplacian reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 110 (2003), 377-383. doi: 10.1023/A:1021083015108. [30] C. Mascia and M. Strani, Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws, SIAM J. Math. Anal., 45 (2013), 3084-3113. doi: 10.1137/120875119. [31] F. Otto and M. G. Reznikoff, Slow motion of gradient flows, J. Differential Equations, 237 (2007), 372-420. doi: 10.1016/j.jde.2007.03.007. [32] A. Petrosyan and E. Valdinoci, Density estimates for a degenerate/singular phase-transition model, SIAM J. Math. Anal., 36 (2005), 1057-1079. doi: 10.1137/S0036141003437678. [33] L. G. Reyna and M. J. Ward, On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math., 48 (1995), 79-120. doi: 10.1002/cpa.3160480202. [34] M. Strani, On the metastable behavior of solutions to a class of parabolic systems, Asymptot. Anal., 90 (2014), 325-344. doi: 10.3233/ASY-141250. [35] S. Takeuchi and Y. Yamada, Asymptotic properties of a reaction-diffusion equation with degenerate$p$-Laplacian, Nonlinear Anal., 42 (2000), 41-61. doi: 10.1016/S0362-546X(98)00329-0. show all references ##### References:  [1] M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2. [2] J. Benedikt, P. Girg, L. Kotrla and P. Takáč, Origin of the$p$-Laplacian and A. Missbach, Electron. J. Differ. Eq., 2018 (2018), 1-17. [3] F. Bethuel and D. Smets, Slow motion for equal depth multiple-well gradient systems: The degenerate case, Discrete Contin. Dyn. Syst., 33 (2013), 67-87. doi: 10.3934/dcds.2013.33.67. [4] L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997. doi: 10.1002/cpa.3160430804. [5] L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237. doi: 10.1016/0022-0396(91)90147-2. [6] J. Carr and R. L. Pego, Metastable patterns in solutions of$u_t = \epsilon^2u_xx-f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576. doi: 10.1002/cpa.3160420502. [7] J. Carr and R. L. Pego, Invariant manifolds for metastable patterns in$u_t = \epsilon^2u_xx-f(u)$, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 133-160. doi: 10.1017/S0308210500031425. [8] M. S. Chang, S. C. Lee and C.-C. Yen, Minimizers and gamma-convergence of energy functionals derived from$p$-Laplacian equation, Taiwanese J. Math., 13 (2009), 2021-2036. doi: 10.11650/twjm/1500405655. [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. [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. [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. [12] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [13] S. Dipierro, A. Farina and E. Valdinoci, Density estimates for degenerate double-well potentials, SIAM J. Math. Anal., 50 (2018), 6333-6347. doi: 10.1137/17M114933X. [14] S. Dipierro, A. 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. [15] L. Evans, H. 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. [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. [17] R. Folino, Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems, Electron. J. Differ. Eq., 2019 (2019), 1-21. [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. [19] R. Folino, C. 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. [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. [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. [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. [23] C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34. doi: 10.1137/S0036141092226053. [24] E. J. Hurtado and M. Sônego, On the energy functionals derived from a non-homogeneous$p$-Laplacian equation:$\Gamma$-convergence, local minimizers and stable transition layers,, J. Math. Anal. Appl., 483 (2020), 123634, 13pp. doi: 10.1016/j.jmaa.2019.123634. [25] J. G. L. Laforgue and R. E. O'Malley Jr., Shock layer movement for Burgers' equation, SIAM J. Appl. Math., 55 (1995), 332-347. doi: 10.1137/S003613999326928X. [26] K. A. Lee, A. Petrosyan and J. L. Vázquez, Large-time geometric properties of solutions of the evolution$p$-Laplacian equation, J. Differential Equations, 229 (2006), 389-411. doi: 10.1016/j.jde.2005.07.028. [27] P. Lindqvist, Notes on the Stationary$p$-Laplace Equation, SpringerBriefs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-14501-9. [28] J. L. Lions, Quelques Méthodes de Résolution des Problèemes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. [29] B. Lou, Singular limit of a$p$-Laplacian reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 110 (2003), 377-383. doi: 10.1023/A:1021083015108. [30] C. Mascia and M. Strani, Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws, SIAM J. Math. Anal., 45 (2013), 3084-3113. doi: 10.1137/120875119. [31] F. Otto and M. G. Reznikoff, Slow motion of gradient flows, J. Differential Equations, 237 (2007), 372-420. doi: 10.1016/j.jde.2007.03.007. [32] A. Petrosyan and E. Valdinoci, Density estimates for a degenerate/singular phase-transition model, SIAM J. Math. Anal., 36 (2005), 1057-1079. doi: 10.1137/S0036141003437678. [33] L. G. Reyna and M. J. Ward, On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math., 48 (1995), 79-120. doi: 10.1002/cpa.3160480202. [34] M. Strani, On the metastable behavior of solutions to a class of parabolic systems, Asymptot. Anal., 90 (2014), 325-344. doi: 10.3233/ASY-141250. [35] S. Takeuchi and Y. Yamada, Asymptotic properties of a reaction-diffusion equation with degenerate$p$-Laplacian, Nonlinear Anal., 42 (2000), 41-61. doi: 10.1016/S0362-546X(98)00329-0. Plots of the potential function (1.4) for$ \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 $) cases, respectively The level of the energy (given by the constant$ 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 $) Solutions to (1.3) for$ \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) $Solutions to (1.3) for$ \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 $Solutions to (1.3) for$ \varepsilon = 0.1 $,$ \theta = 8 $and$ p = \pi $(left) and$ p = 5.5 $(right); the initial datum$ u_0 $is as in Figure 3  [1] L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with$p\& q$-Laplacian. Communications on Pure and Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9 [2] Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587 [3] Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure and Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371 [4] Shin-Ichiro Ei, Hiroshi Matsuzawa. The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 901-921. doi: 10.3934/dcds.2010.26.901 [5] Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171 [6] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922 [7] Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [8] Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Dead cores and bursts for p-Laplacian elliptic equations with weights. Conference Publications, 2007, 2007 (Special) : 191-200. doi: 10.3934/proc.2007.2007.191 [9] Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593 [10] Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2117-2138. doi: 10.3934/cpaa.2021060 [11] Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 [12] Dimitri Mugnai, Kanishka Perera, Edoardo Proietti Lippi. A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications. Communications on Pure and Applied Analysis, 2022, 21 (1) : 275-292. doi: 10.3934/cpaa.2021177 [13] Nikolaos S. Papageorgiou, Vicenţiu D. Rǎdulescu, Dušan D. Repovš. Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term. Communications on Pure and Applied Analysis, 2018, 17 (1) : 231-241. doi: 10.3934/cpaa.2018014 [14] Maurizio Garrione, Marta Strani. Monotone wave fronts for$(p, q)\$-Laplacian driven reaction-diffusion equations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 91-103. doi: 10.3934/dcdss.2019006 [15] Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 [16] Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020 [17] Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130 [18] Annegret Glitzky. Energy estimates for electro-reaction-diffusion systems with partly fast kinetics. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 159-174. doi: 10.3934/dcds.2009.25.159 [19] Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254 [20] Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure and Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475

2021 Impact Factor: 1.588