# American Institute of Mathematical Sciences

December  2021, 41(12): 6023-6046. doi: 10.3934/dcds.2021105

## Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivity

 1 Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Reggio Emilia, 42122, Italy 2 Department of Mathematics and Computer Science, University of Ferrara, Ferrara, 44121, Italy

* Corresponding author: Andrea Corli

Received  November 2020 Revised  April 2021 Published  December 2021 Early access  June 2021

We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or even more than once; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviors at the points where the diffusivity degenerates. In particular, if such points are interior points, then the sharp behaviors are new and unusual.

Citation: Diego Berti, Andrea Corli, Luisa Malaguti. Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivity. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 6023-6046. doi: 10.3934/dcds.2021105
##### References:
 [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5. [2] L. Bao and Z. Zhou, Traveling wave in backward and forward parabolic equations from population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1507-1522.  doi: 10.3934/dcdsb.2014.19.1507. [3] L. Bao and Z. Zhou, Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 395-412.  doi: 10.3934/dcdss.2017019. [4] N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow. I. Fluid dynamic and kinetic modelling, Math. Models Methods Appl. Sci., 12 (2002), 1801-1843.  doi: 10.1142/S0218202502002343. [5] N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463.  doi: 10.1137/090746677. [6] D. Berti, A. Corli and L. Malaguti, Uniqueness and nonuniqueness of fronts for degenerate diffusion-convection reaction equations, Electron. J. Qual. Theory Differ. Equ., (2020), Paper No. 66, 34 pp. doi: 10.14232/ejqtde.2020.1.66. [7] L. Bruno, A. Tosin, P. Tricerri and F. Venuti, Non-local first-order modelling of crowd dynamics: A multidimensional framework with applications, Appl. Math. Model., 35 (2011), 426-445.  doi: 10.1016/j.apm.2010.07.007. [8] A. Corli, L. di Ruvo and L. Malaguti, Sharp profiles in models of collective movements, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 40, 31 pp. doi: 10.1007/s00030-017-0460-z. [9] A. Corli, L. di Ruvo, L. Malaguti and M. D. Rosini, Traveling waves for degenerate diffusive equations on networks, Netw. Heterog. Media, 12 (2017), 339-370.  doi: 10.3934/nhm.2017015. [10] A. Corli and L. Malaguti, Semi-wavefront solutions in models of collective movements with density-dependent diffusivity, Dyn. Partial Differ. Equ., 13 (2016), 297-331.  doi: 10.4310/DPDE.2016.v13.n4.a2. [11] A. Corli and L. Malaguti, Viscous profiles in models of collective movement with negative diffusivity, Z. Angew. Math. Phys., 70 (2019), Art. 47, 22 pp. doi: 10.1007/s00033-019-1094-2. [12] A. Corli and L. Malaguti, Wavefronts in traffic flows and crowds dynamics, in Anomalies in Partial Differential Equations (eds. M. Cicognani, D. Del Santo, A. Parmeggiani, M. Reissig), Springer Indam Series, 43 (2021), 167–189. [13] D. A. DiCarlo, R. Juanes, L. Tara and T. P. Witelski, Nonmonotonic traveling wave solutions of infiltration into porous media, Water Resources Res., 44 (2008), 1-12. [14] L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation models, Adv. Dyn. Syst. Appl., 4 (2009), 19-33. [15] M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, 2016. [16] M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. [17] B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4. [18] D. Horstmann, K. J. Painter and H. G. Othmer, Aggregation under local reinforcement: From lattice to continuum, European J. Appl. Math., 15 (2004), 546-576.  doi: 10.1017/S0956792504005571. [19] B. S. Kerner and V. V. Osipov, Autosolitons, Kluwer Academic Publishers Group, Dordrecht, 1994. doi: 10.1007/978-94-017-0825-8. [20] M. Kuzmin and S. Ruggerini, Front propagation in diffusion-aggregation models with bi-stable reaction, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 819-833.  doi: 10.3934/dcdsb.2011.16.819. [21] P. K. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1175-1189.  doi: 10.3934/dcdsb.2006.6.1175. [22] P. K. Maini, L. Malaguti, C. Marcelli and S. Matucci, Aggregative movement and front propagation for bi-stable population models, Math. Models Methods Appl. Sci., 17 (2007), 1351-1368.  doi: 10.1142/S0218202507002303. [23] L. Malaguti and C. Marcelli, Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Differential Equations, 195 (2003), 471-496.  doi: 10.1016/j.jde.2003.06.005. [24] L. Malaguti and C. Marcelli, Finite speed of propagation in monostable degenerate reaction-diffusion-convection equations, Adv. Nonlinear Stud., 5 (2005), 223-252.  doi: 10.1515/ans-2005-0204. [25] C. Marcelli and F. Papalini, A new estimate of the minimal wave speed for travelling fronts in reaction-diffusion-convection equations, Electron. J. Qual. Theory Differ. Equ., (2018), Paper No. 10, 13 pp. doi: 10.14232/ejqtde.2018.1.10. [26] P. Nelson, Synchronized traffic flow from a modified {L}ighthill-{W}hitham model, Phys. Review E, 61 (2000), R6052–R6055. [27] V. Padrón, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3. [28] M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Springer, Heidelberg, 2013. doi: 10.1007/978-3-319-00155-5.

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##### References:
 [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5. [2] L. Bao and Z. Zhou, Traveling wave in backward and forward parabolic equations from population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1507-1522.  doi: 10.3934/dcdsb.2014.19.1507. [3] L. Bao and Z. Zhou, Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 395-412.  doi: 10.3934/dcdss.2017019. [4] N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow. I. Fluid dynamic and kinetic modelling, Math. Models Methods Appl. Sci., 12 (2002), 1801-1843.  doi: 10.1142/S0218202502002343. [5] N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463.  doi: 10.1137/090746677. [6] D. Berti, A. Corli and L. Malaguti, Uniqueness and nonuniqueness of fronts for degenerate diffusion-convection reaction equations, Electron. J. Qual. Theory Differ. Equ., (2020), Paper No. 66, 34 pp. doi: 10.14232/ejqtde.2020.1.66. [7] L. Bruno, A. Tosin, P. Tricerri and F. Venuti, Non-local first-order modelling of crowd dynamics: A multidimensional framework with applications, Appl. Math. Model., 35 (2011), 426-445.  doi: 10.1016/j.apm.2010.07.007. [8] A. Corli, L. di Ruvo and L. Malaguti, Sharp profiles in models of collective movements, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 40, 31 pp. doi: 10.1007/s00030-017-0460-z. [9] A. Corli, L. di Ruvo, L. Malaguti and M. D. Rosini, Traveling waves for degenerate diffusive equations on networks, Netw. Heterog. Media, 12 (2017), 339-370.  doi: 10.3934/nhm.2017015. [10] A. Corli and L. Malaguti, Semi-wavefront solutions in models of collective movements with density-dependent diffusivity, Dyn. Partial Differ. Equ., 13 (2016), 297-331.  doi: 10.4310/DPDE.2016.v13.n4.a2. [11] A. Corli and L. Malaguti, Viscous profiles in models of collective movement with negative diffusivity, Z. Angew. Math. Phys., 70 (2019), Art. 47, 22 pp. doi: 10.1007/s00033-019-1094-2. [12] A. Corli and L. Malaguti, Wavefronts in traffic flows and crowds dynamics, in Anomalies in Partial Differential Equations (eds. M. Cicognani, D. Del Santo, A. Parmeggiani, M. Reissig), Springer Indam Series, 43 (2021), 167–189. [13] D. A. DiCarlo, R. Juanes, L. Tara and T. P. Witelski, Nonmonotonic traveling wave solutions of infiltration into porous media, Water Resources Res., 44 (2008), 1-12. [14] L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation models, Adv. Dyn. Syst. Appl., 4 (2009), 19-33. [15] M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, 2016. [16] M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. [17] B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4. [18] D. Horstmann, K. J. Painter and H. G. Othmer, Aggregation under local reinforcement: From lattice to continuum, European J. Appl. Math., 15 (2004), 546-576.  doi: 10.1017/S0956792504005571. [19] B. S. Kerner and V. V. Osipov, Autosolitons, Kluwer Academic Publishers Group, Dordrecht, 1994. doi: 10.1007/978-94-017-0825-8. [20] M. Kuzmin and S. Ruggerini, Front propagation in diffusion-aggregation models with bi-stable reaction, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 819-833.  doi: 10.3934/dcdsb.2011.16.819. [21] P. K. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1175-1189.  doi: 10.3934/dcdsb.2006.6.1175. [22] P. K. Maini, L. Malaguti, C. Marcelli and S. Matucci, Aggregative movement and front propagation for bi-stable population models, Math. Models Methods Appl. Sci., 17 (2007), 1351-1368.  doi: 10.1142/S0218202507002303. [23] L. Malaguti and C. Marcelli, Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Differential Equations, 195 (2003), 471-496.  doi: 10.1016/j.jde.2003.06.005. [24] L. Malaguti and C. Marcelli, Finite speed of propagation in monostable degenerate reaction-diffusion-convection equations, Adv. Nonlinear Stud., 5 (2005), 223-252.  doi: 10.1515/ans-2005-0204. [25] C. Marcelli and F. Papalini, A new estimate of the minimal wave speed for travelling fronts in reaction-diffusion-convection equations, Electron. J. Qual. Theory Differ. Equ., (2018), Paper No. 10, 13 pp. doi: 10.14232/ejqtde.2018.1.10. [26] P. Nelson, Synchronized traffic flow from a modified {L}ighthill-{W}hitham model, Phys. Review E, 61 (2000), R6052–R6055. [27] V. Padrón, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3. [28] M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Springer, Heidelberg, 2013. doi: 10.1007/978-3-319-00155-5.
Typical plots of the functions $g$ and $D$.
The thresholds $c^*_{p,r}$, $c^*_{n,l}$ used in (2.7) and $c^*_{n,r}$, $c^*_{p,l}$, used below in (2.15).
Some possible wavefronts joining $1$ with $0$ in case $\rm (D_{pn})$: a classical wavefront $\varphi^1$ (for $c>c_{pn}^*$ and either $\dot D(\alpha)<0$ or $c>h(\alpha)$); a wavefront $\varphi^2$ which is sharp at $1$, with finite right derivative at $\xi_1^2$ and $( \varphi^2)'(\xi_\alpha^2) = -\infty$ (for $c = c_{n,l}^*>h(1)$, $D(1) = 0$, $\dot D(1)>0$, $\dot D(\alpha) = 0$ and $c\le h(\alpha)$); a wavefront $\varphi^3$ which is sharp at $0$ with $( \varphi^3)'(\xi_0^3) = -\infty$ (for $c = c_{p,r}^*> h(0)$, $D(0) = 0 = \dot D(0)$).
$D$ satisfies $(\rm D_{pn})$: Left: the plots of $D$, $g$ in $[0,\alpha]$ (solid lines) and in $[\alpha,1]$ (dashed lines); right, a corresponding profile.
Construction of the profile $\varphi$ in the case $\xi_0,\xi_1\in \mathbb R$.
$D$ satisfies $(\rm D_{np})$. Left: the plots of $D$, $g$ in $[0,\beta]$ (dashed lines) and $[\beta,1]$ (solid lines); right, a corresponding profile.
Some possible wavefronts joining $1$ with $0$ in case $\rm (D_{np})$; Profiles are labelled according to the cases (1)–(4) of Remark 6; $\varphi^5$ occurs in both cases (3) and (4). For simplicity we only represented strictly monotone profiles.
Typical plots of the functions $D$.
The pasting of the profiles.
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