Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations

Maurizio Garrione; Marta Strani

doi:10.3934/dcdss.2019006

Article Contents

2019, Volume 12, Issue 1: 91-103. Doi: 10.3934/dcdss.2019006

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Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations

Maurizio Garrione^1, and
Marta Strani^2,

1.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy
2.
Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca' Foscari, Via Torino 155, Venezia Mestre, 30172, Italy

Received: March 2017

Revised: September 2017

Published online: July 31, 2018

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Abstract

We study the existence of monotone heteroclinic traveling waves for the $-dimensional reaction-diffusion equation

$u_t = (\vert u_x \vert^{p-2} u_x + \vert u_x \vert^{q-2} u_x)_x + f(u), \;\;\;\; t ∈ \mathbb{R}, \; x ∈ \mathbb{R}, $

where the non-homogeneous operator appearing on the right-hand side is of $(p, q)$-Laplacian type. Here we assume that $2 ≤ q < p$ and $f$ is a nonlinearity of Fisher type on $[0, 1]$, namely $f(0) = 0 = f(1)$ and $f > 0$ on $]0, 1[\, $. We give an estimate of the critical speed and we comment on the roles of $p$ and $q$ in the dynamics, providing some numerical simulations.

Keywords:

Mathematics Subject Classification: Primary: 35K55, 35K57; Secondary: 35J92, 34C37.

Citation:

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Figure 1. The graph of $\mathcal{G}_c(\beta)$ for $p = 4$, $q = 3$, $L_+ = 6$ and $c$ given by 9$_{(i)}$

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Figure 2. The solution of 25 with $p = 4$, $q = 2$, $H = 1$, and $f(v) = v^{q'-1} (1-v)$, for $c = 2$. Actually, with the notations used throughout the section, here $L = 1$, $q = q' = 2$, so the lower bound for $c^*$ is equal to $2$ and it actually seems that $c = 2$ is already an admissible speed. Notice that $y$ is indeed very small, so the process appears ruled by the $q$-Laplacian

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Figure 3. The solution of the differential equation in 25 satisfying $y(H) = 0$, with $p = 4$, $q = 2$, $H = 7$, $f(v) = v^{q'-1} (H-v)$ and $c = 2\sqrt{7}$; here $L = 7$ and the bound 27 would give $c^* \geq 2\sqrt{7}$. However, this speed appears far from being admissible and, indeed, $y$ reaches values much greater than $1$, for which the $p$-Laplacian rules the diffusion

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Figure 4. The solution of 25 with $p = 4$, $q = 2$, $Q(s) = \tfrac{q-1}{q} \vert s \vert^q - \tfrac{p-1}{p} \vert s \vert^p$, $f(v) = v^{q'-1} (1-v)$ and $c = 2$; here $y$ stays well below the threshold of invertibility of $Q$, so that the situation roughly appears the same as in Figure 2 and the bound given by 12 already gives an admissible speed

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Figure 5. The solution of the differential equation in 25 satisfying $y(H) = 0$, for $H = 4$, $f(v) = v^{q'-1} (4-v)$ and the other positions as in Figure 4. Here the derivative of the wave profile has to increase too much in order for $y$ to connect the equilibria $0$ and $4$, so that we enter the non-invertibility region for $Q$. Thus, we are not able to find any solution of 5

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