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

Raffaele Folino; Ramón G. Plaza; Marta Strani

doi:10.3934/dcds.2020403

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2021, Volume 41, Issue 7: 3211-3240. Doi: 10.3934/dcds.2020403

This issue Previous Article Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice Next Article Another point of view on Kusuoka's measure

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
^* Corresponding author

Received: April 30, 2020

Revised: September 30, 2020

Early access: December 2020

Published: July 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K91, 35K57; Secondary: 35B36, 35B40.

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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

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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 $)

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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) $

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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 $

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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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