Traveling wave in backward and forward parabolic equations from population dynamics

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

Traveling wave in backward and forward parabolic equations from population dynamics

Lianzhang Bao ^1, and Zhengfang Zhou ^2,

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

Abstract
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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 $u\equiv 1$.

Keywords: Aggregation diffusion processes, population dynamics, traveling wave solution..

Mathematics Subject Classification: Primary: 35K57, 92D25; Secondary: 34B1.

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.

show all references

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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