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February  2019, 12(1): 91-103. doi: 10.3934/dcdss.2019006

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  July 2018

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: Reaction-diffusion equations, $(p, q)$-Laplacian, wave fronts, admissible speeds.
Mathematics Subject Classification: Primary: 35K55, 35K57; Secondary: 35J92, 34C37.
Citation: Maurizio Garrione, Marta Strani. Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 91-103. doi: 10.3934/dcdss.2019006
References:
[1]

A. Audrito and J. L. Vazquez, The Fisher-KPP problem with doubly nonlinear diffusion, J. Differential Equations, 263 (2017), 7647-7708.  doi: 10.1016/j.jde.2017.08.025.  Google Scholar

[2]

R. Bartolo, A. M. Candela and A. Salvatore, On a class of superlinear $(p, q)$-Laplacian type equations on $\mathbb{R}^N$, J. Math. Anal. Appl., 438 (2016), 29-41.  doi: 10.1016/j.jmaa.2016.01.049.  Google Scholar

[3]

V. Benci, D. Fortunato and L. Pisani, Soliton like solutions of a Lorentz invariant equation in dimension $3$, Rev. Math. Phys., 10 (1998), 315-344.  doi: 10.1142/S0129055X98000100.  Google Scholar

[4]

D. Bonheure and L. Sanchez, Heteroclinic orbits for some classes of second and fourth order differential equations, in Handbook of Differential Equations: Ordinary Differential Equations, (eds. A. Canada, P. Drábek and A. Fonda), Elsevier, Amsterdam, 3 (2006), 103–202. doi: 10.1016/S1874-5725(06)80006-4.  Google Scholar

[5]

L. Cherfils and Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with $p\And q$-Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22.  doi: 10.3934/cpaa.2005.4.9.  Google Scholar

[6]

I. Coelho and L. Sanchez, Traveling wave profiles in some models with nonlinear diffusion, Appl. Math. Comput., 235 (2014), 469-481.  doi: 10.1016/j.amc.2014.02.104.  Google Scholar

[7]

R. Enguiça, A. Gavioli and L. Sanchez, A class of singular first order differential equations with applications in reaction-diffusion, Discrete Contin. Dyn. Syst., 33 (2013), 173-191.  doi: 10.3934/dcds.2013.33.173.  Google Scholar

[8]

E. Feireisl, D. Hilhorst, H. Petzeltová and P. Takáč, Front propagation in nonlinear parabolic equations, J. London Math. Soc., 90 (2014), 551-572.  doi: 10.1112/jlms/jdu039.  Google Scholar

[9]

R. A. Fisher, The wave of advance of advantageous genus, Ann. Eugenics, 7 (1937), 355-369.   Google Scholar

[10]

M. Garrione and L. Sanchez, Monotone traveling waves for reaction-diffusion equations involving the curvature operator, Boundary Value Probl., 2015 (2015), 31 pp. doi: 10.1186/s13661-015-0303-y.  Google Scholar

[11]

M. Garrione and M. Strani, Heteroclinic traveling fronts for a generalized Fisher-Burgers equation with saturating diffusion, preprint, arXiv: 1702.03782. Google Scholar

[12]

A. Gavioli and L. Sanchez, A variational property of critical speed to travelling waves in the presence of nonlinear diffusion, Appl. Math. Lett., 48 (2015), 47-54.  doi: 10.1016/j.aml.2015.03.011.  Google Scholar

[13]

C. J. He and G. B. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing $p\And q$-Laplacians, Ann. Acad. Sci. Fenn. Math., 33 (2008), 337-371.   Google Scholar

[14]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, (French) [Study of the diffusion equation with growth of the matter quantity and applications to a biological problem], Mosc. Univ. Math. Bull., 1 (1937), 1–25. Google Scholar

[15]

Y. Yang and K. Perera, $(N, q)$-Laplacian problems with critical Trudinger-Moser nonlinearities, Bull. London Math. Soc., 48 (2016), 260-270.  doi: 10.1112/blms/bdw002.  Google Scholar

show all references

References:
[1]

A. Audrito and J. L. Vazquez, The Fisher-KPP problem with doubly nonlinear diffusion, J. Differential Equations, 263 (2017), 7647-7708.  doi: 10.1016/j.jde.2017.08.025.  Google Scholar

[2]

R. Bartolo, A. M. Candela and A. Salvatore, On a class of superlinear $(p, q)$-Laplacian type equations on $\mathbb{R}^N$, J. Math. Anal. Appl., 438 (2016), 29-41.  doi: 10.1016/j.jmaa.2016.01.049.  Google Scholar

[3]

V. Benci, D. Fortunato and L. Pisani, Soliton like solutions of a Lorentz invariant equation in dimension $3$, Rev. Math. Phys., 10 (1998), 315-344.  doi: 10.1142/S0129055X98000100.  Google Scholar

[4]

D. Bonheure and L. Sanchez, Heteroclinic orbits for some classes of second and fourth order differential equations, in Handbook of Differential Equations: Ordinary Differential Equations, (eds. A. Canada, P. Drábek and A. Fonda), Elsevier, Amsterdam, 3 (2006), 103–202. doi: 10.1016/S1874-5725(06)80006-4.  Google Scholar

[5]

L. Cherfils and Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with $p\And q$-Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22.  doi: 10.3934/cpaa.2005.4.9.  Google Scholar

[6]

I. Coelho and L. Sanchez, Traveling wave profiles in some models with nonlinear diffusion, Appl. Math. Comput., 235 (2014), 469-481.  doi: 10.1016/j.amc.2014.02.104.  Google Scholar

[7]

R. Enguiça, A. Gavioli and L. Sanchez, A class of singular first order differential equations with applications in reaction-diffusion, Discrete Contin. Dyn. Syst., 33 (2013), 173-191.  doi: 10.3934/dcds.2013.33.173.  Google Scholar

[8]

E. Feireisl, D. Hilhorst, H. Petzeltová and P. Takáč, Front propagation in nonlinear parabolic equations, J. London Math. Soc., 90 (2014), 551-572.  doi: 10.1112/jlms/jdu039.  Google Scholar

[9]

R. A. Fisher, The wave of advance of advantageous genus, Ann. Eugenics, 7 (1937), 355-369.   Google Scholar

[10]

M. Garrione and L. Sanchez, Monotone traveling waves for reaction-diffusion equations involving the curvature operator, Boundary Value Probl., 2015 (2015), 31 pp. doi: 10.1186/s13661-015-0303-y.  Google Scholar

[11]

M. Garrione and M. Strani, Heteroclinic traveling fronts for a generalized Fisher-Burgers equation with saturating diffusion, preprint, arXiv: 1702.03782. Google Scholar

[12]

A. Gavioli and L. Sanchez, A variational property of critical speed to travelling waves in the presence of nonlinear diffusion, Appl. Math. Lett., 48 (2015), 47-54.  doi: 10.1016/j.aml.2015.03.011.  Google Scholar

[13]

C. J. He and G. B. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing $p\And q$-Laplacians, Ann. Acad. Sci. Fenn. Math., 33 (2008), 337-371.   Google Scholar

[14]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, (French) [Study of the diffusion equation with growth of the matter quantity and applications to a biological problem], Mosc. Univ. Math. Bull., 1 (1937), 1–25. Google Scholar

[15]

Y. Yang and K. Perera, $(N, q)$-Laplacian problems with critical Trudinger-Moser nonlinearities, Bull. London Math. Soc., 48 (2016), 260-270.  doi: 10.1112/blms/bdw002.  Google Scholar

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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