# American Institute of Mathematical Sciences

August  2014, 19(6): 1507-1522. doi: 10.3934/dcdsb.2014.19.1507

## Traveling wave in backward and forward parabolic equations from population dynamics

 1 College of Mathematics, Jilin University, Changchun, Jilin 130012, China 2 Department of Mathematics, Michigan State University, East Lansing, MI 48824

Received  October 2013 Revised  February 2014 Published  June 2014

This work is concerned with the properties of the traveling wave of the backward and forward parabolic equation \begin{equation*} u_t= [ D(u)u_x]_x + g(u),\quad t\geq 0, x\in \mathbb{R}, \end{equation*} where $D(u)$ changes its sign once, from negative to positive value, in the interval $u\in [0,1]$ and $g(u)$ is a mono-stable nonlinear reaction term. The existence of infinitely many traveling wave solutions is proven. These traveling waves are parameterized by their wave speed and monotonically connect the stationary states $u\equiv0$ and $u\equiv 1$.
Citation: Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507
##### References:
 [1] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Traveling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002. [2] L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation models, Advances in Dynamical Systems and Applications, 4 (2009), 19-33. [3] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. [4] F. S. Garduño and P. K. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192. doi: 10.1007/BF00160178. [5] F. S. Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equation, 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055. [6] F. S. Garduño, P. K. Maini and J. Pérez-Velázquez, A non-linear degenerate equation for direct aggregation and taravelling wave dynamics, Discrete and Continuous Dynamical Systerms Series B, 13 (2010), 455-487. doi: 10.3934/dcdsb.2010.13.455. [7] K. P. Hadeler, Travelling fronts and free boundary value problems， In Albretch, J. Collatz, L. Hoffman, K. H. (eds.) Numerical Treatment of Free Boundary Value Problems. Basel: Birkhauser, 1981. [8] D. Horstmann, K. J. Painter and H. G. Othmer, Aggregation under local reinforcement: From lattice to continuum, Euro. Jnl of Applied Mathematics, 15 (2004), 546-576. doi: 10.1017/S0956792504005571. [9] A. Kolmogorov, I. Petrovsky and I. N. Piskounov, Study of the diffusion equation with growth of the quantity of matter and its applications to a biological problem， (English translation containing the relvent results) In OLiveira-Pinto, F., Conolly, B.W.(eds.) Applicable mathematics of non-physical phenomena}. New York: Wiley, 1982. [10] M. Kuzmin and S. Ruggerini, Front Propagation in Diffusion-Aggregation Models with Bi-Stable Reaction, Discrete and Continuous Dynamical Systems Series B, 16 (2011), 819-833. doi: 10.3934/dcdsb.2011.16.819. [11] T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Diff. Eqns., 32 (2007), 1941-1964. doi: 10.1080/03605300701318955. [12] D. Li and X. Zhang, On a nonlocal aggregation model with nolinear diffusion, Discrete and Continuous Dynamical Systems, 27 (2010), 301-323. doi: 10.3934/dcds.2010.27.301. [13] P. K. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete and Continuous Dynamical Systerms Series B, 6 (2006), 1175-1189. doi: 10.3934/dcdsb.2006.6.1175. [14] L. Malaguti and C. Marcelli, Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms, Math. Nachr., 242 (2002), 148-164. doi: 10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J. [15] L. Malaguti and C. Marcelli, Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Differential Equation, 195 (2003), 471-496. doi: 10.1016/j.jde.2003.06.005. [16] V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Diff. Eqns., 23 (1998), 457-486. doi: 10.1080/03605309808821353. [17] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.

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##### References:
 [1] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Traveling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002. [2] L. Ferracuti, C. Marcelli and F. Papalini, Travelling waves in some reaction-diffusion-aggregation models, Advances in Dynamical Systems and Applications, 4 (2009), 19-33. [3] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. [4] F. S. Garduño and P. K. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192. doi: 10.1007/BF00160178. [5] F. S. Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equation, 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055. [6] F. S. Garduño, P. K. Maini and J. Pérez-Velázquez, A non-linear degenerate equation for direct aggregation and taravelling wave dynamics, Discrete and Continuous Dynamical Systerms Series B, 13 (2010), 455-487. doi: 10.3934/dcdsb.2010.13.455. [7] K. P. Hadeler, Travelling fronts and free boundary value problems， In Albretch, J. Collatz, L. Hoffman, K. H. (eds.) Numerical Treatment of Free Boundary Value Problems. Basel: Birkhauser, 1981. [8] D. Horstmann, K. J. Painter and H. G. Othmer, Aggregation under local reinforcement: From lattice to continuum, Euro. Jnl of Applied Mathematics, 15 (2004), 546-576. doi: 10.1017/S0956792504005571. [9] A. Kolmogorov, I. Petrovsky and I. N. Piskounov, Study of the diffusion equation with growth of the quantity of matter and its applications to a biological problem， (English translation containing the relvent results) In OLiveira-Pinto, F., Conolly, B.W.(eds.) Applicable mathematics of non-physical phenomena}. New York: Wiley, 1982. [10] M. Kuzmin and S. Ruggerini, Front Propagation in Diffusion-Aggregation Models with Bi-Stable Reaction, Discrete and Continuous Dynamical Systems Series B, 16 (2011), 819-833. doi: 10.3934/dcdsb.2011.16.819. [11] T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Diff. Eqns., 32 (2007), 1941-1964. doi: 10.1080/03605300701318955. [12] D. Li and X. Zhang, On a nonlocal aggregation model with nolinear diffusion, Discrete and Continuous Dynamical Systems, 27 (2010), 301-323. doi: 10.3934/dcds.2010.27.301. [13] P. K. Maini, L. Malaguti, C. Marcelli and S. Matucci, Diffusion-aggregation processes with mono-stable reaction terms, Discrete and Continuous Dynamical Systerms Series B, 6 (2006), 1175-1189. doi: 10.3934/dcdsb.2006.6.1175. [14] L. Malaguti and C. Marcelli, Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms, Math. Nachr., 242 (2002), 148-164. doi: 10.1002/1522-2616(200207)242:1<148::AID-MANA148>3.0.CO;2-J. [15] L. Malaguti and C. Marcelli, Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations, J. Differential Equation, 195 (2003), 471-496. doi: 10.1016/j.jde.2003.06.005. [16] V. Padrón, Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Partial Diff. Eqns., 23 (1998), 457-486. doi: 10.1080/03605309808821353. [17] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218. doi: 10.1093/biomet/38.1-2.196.
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