# American Institute of Mathematical Sciences

• Previous Article
Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces
• DCDS-S Home
• This Issue
• Next Article
Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains
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

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. Citation: 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 ##### 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. [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. [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. [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. [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. [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. [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. The graph of$\mathcal{G}_c(\beta)$for$p = 4$,$q = 3$,$L_+ = 6$and$c$given by 9$_{(i)}$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 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 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 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  [1] Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407 [2] 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 [3] Antoine Mellet, Jean-Michel Roquejoffre, Yannick Sire. Generalized fronts for one-dimensional reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 303-312. doi: 10.3934/dcds.2010.26.303 [4] Masaharu Taniguchi. Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 1011-1046. doi: 10.3934/dcds.2012.32.1011 [5] Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126 [6] Masaharu Taniguchi. Traveling fronts in perturbed multistable reaction-diffusion equations. Conference Publications, 2011, 2011 (Special) : 1368-1377. doi: 10.3934/proc.2011.2011.1368 [7] Henri Berestycki, Guillemette Chapuisat. Traveling fronts guided by the environment for reaction-diffusion equations. Networks and Heterogeneous Media, 2013, 8 (1) : 79-114. doi: 10.3934/nhm.2013.8.79 [8] Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267 [9] Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709 [10] Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001 [11] Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115 [12] Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147 [13] 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 [14] Hans F. Weinberger, Kohkichi Kawasaki, Nanako Shigesada. Spreading speeds for a partially cooperative 2-species reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1087-1098. doi: 10.3934/dcds.2009.23.1087 [15] Yacheng Liu, Runzhang Xu. Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 171-189. doi: 10.3934/dcdsb.2007.7.171 [16] Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681 [17] Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations and Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43 [18] Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281 [19] Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083 [20] Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations and Control Theory, 2021, 10 (4) : 701-722. doi: 10.3934/eect.2020087

2021 Impact Factor: 1.865