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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February
2019, 12(1): 91-103.
doi: 10.3934/dcdss.2019006
Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations
| 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 |
We study the existence of monotone heteroclinic traveling waves for the
| $ |
| $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
| $(p, q)$ |
| $2 ≤ q < p$ |
| $f$ |
| $[0, 1]$ |
| $f(0) = 0 = f(1)$ |
| $f > 0$ |
| $]0, 1[\, $ |
| $p$ |
| $q$ |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
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
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
Figure 2 and the bound given by 12 already gives an admissible speed">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
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">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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