A class of singular first order differential equations with applications in reaction-diffusion

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2013, 33(1): 173-191. doi: 10.3934/dcds.2013.33.173

A class of singular first order differential equations with applications in reaction-diffusion

Ricardo Enguiça ^1, , Andrea Gavioli ^2, and Luís Sanchez ^3,

1.	Area Departamental de Matemática, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro, 1 - 1950-062 Lisboa, Portugal
2.	Dipartimento di Matematica Pura ed Applicata, Univ. di Modena e Reggio Emilia, Via Campi, 213b, 41100 Modena, Italy
3.	Faculdade de Ciências da Universidade de Lisboa, CMAF, Avenida Professor Gama Pinto 2, 1649-003 Lisboa, Portugal

Received August 2011 Revised February 2012 Published September 2012

Abstract
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We study positive solutions $y(u)$ for the first order differential equation $$y'=q(c\,{y}^{\frac{1}{p}}-f(u))$$ where $c>0$ is a parameter, $p>1$ and $q>1$ are conjugate numbers and $f$ is a continuous function in $[0,1]$ such that $f(0)=0=f(1)$. We shall be particularly concerned with positive solutions $y(u)$ such that $y(0)=0=y(1)$. Our motivation lies in the fact that this problem provides a model for the existence of travelling wave solutions for analogues of the FKPP equation in one space dimension, where diffusion is represented by the $p$-Laplacian operator. We obtain a theory of admissible velocities and some other features that generalize classical and recent results, established for $p=2$.

Keywords: $p$-Laplacian, heteroclinic, travelling wave, sharp solution., FKPP equation, critical speed.

Mathematics Subject Classification: Primary: 34B18, 34C37, 35K5.

Citation: Ricardo Enguiça, Andrea Gavioli, Luís Sanchez. A class of singular first order differential equations with applications in reaction-diffusion. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 173-191. doi: 10.3934/dcds.2013.33.173

References:

[1]	M. Arias, J. Campos and C. Marcelli, Fastness and continuous dependence in front propagation in Fisher-KPP equations,, Discrete and Continuous Dynamical Systems Series B, 11 (2009), 11.
[2]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5.
[3]	H. Berestycki and L. Nirenberg, Travelling fronts in cylinders,, Annales de l'Institut Henri Poincare- Analyse non lineaire, 9 (1992), 497.
[4]	D. Bonheure and L. Sanchez, "Heteroclinic Orbits For Some Classes of Second and Fourth Order Differential Equations,'', Handbook of Differential Equations: Ordinary Differential Equations, 3 (2006). doi: 10.1016/S1874-5725(06)80006-4.
[5]	B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction,'', Progress in Nonlinear Differential Equations and their Applications, (2004).
[6]	A. Hamydy, Travelling wave for absorption-convection-diffusion equations,, Electronic Journal of Diff. Eq., 2006 (2006), 1.
[7]	A. Sánchez-Valdés and B. Hernández-Bermejo, New travelling wave solutions for the Fisher-KPP equation with general exponents,, Appl. Math. Lett., 18 (2005), 1281. doi: 10.1016/j.aml.2005.02.016.
[8]	X. Hou, Y. Li and K. Meyer, Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities,, Discrete and Contininuous Dynamical Systems, 26 (2010), 265.
[9]	A. Kolmogorov, I. Petrovski and N. Piscounov, Etude de l'équation de la diffusion avec croissance de la quantité de matiére et son application à un probléme biologique,, Bull. Univ. Moskou Ser. Internat. Sec. A, 1 (1937), 1.
[10]	P. 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. doi: 10.3934/dcdsb.2006.6.1175.
[11]	P. 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. doi: 10.1142/S0218202507002303.
[12]	F. Sánchez-Garduño, P. Maini and J. Pérez-Velázquez, A non-linear degenerate equation for direct aggregation and traveling wave dynamics,, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 455.
[13]	F. Sánchez-Garduño and P. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations,, Journal of Mathematical Biology, 33 (1994), 163. doi: 10.1007/BF00160178.
[14]	L. Malaguti and C. Marcelli, Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms,, Math. Nachr., 242 (2002), 148. 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,, Journal of Differential Equations, 195 (2003), 471. doi: 10.1016/j.jde.2003.06.005.
[16]	P. Pang, Y. Wang and J. Yin, Periodic solutions for a class od reaction-diffusion equations with $p$-Laplacian,, Nonlinear Analysis: Real World Applications, 11 (2010), 323. doi: 10.1016/j.nonrwa.2008.11.006.

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References:

[1]	M. Arias, J. Campos and C. Marcelli, Fastness and continuous dependence in front propagation in Fisher-KPP equations,, Discrete and Continuous Dynamical Systems Series B, 11 (2009), 11.
[2]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, Adv. in Math., 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5.
[3]	H. Berestycki and L. Nirenberg, Travelling fronts in cylinders,, Annales de l'Institut Henri Poincare- Analyse non lineaire, 9 (1992), 497.
[4]	D. Bonheure and L. Sanchez, "Heteroclinic Orbits For Some Classes of Second and Fourth Order Differential Equations,'', Handbook of Differential Equations: Ordinary Differential Equations, 3 (2006). doi: 10.1016/S1874-5725(06)80006-4.
[5]	B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction,'', Progress in Nonlinear Differential Equations and their Applications, (2004).
[6]	A. Hamydy, Travelling wave for absorption-convection-diffusion equations,, Electronic Journal of Diff. Eq., 2006 (2006), 1.
[7]	A. Sánchez-Valdés and B. Hernández-Bermejo, New travelling wave solutions for the Fisher-KPP equation with general exponents,, Appl. Math. Lett., 18 (2005), 1281. doi: 10.1016/j.aml.2005.02.016.
[8]	X. Hou, Y. Li and K. Meyer, Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities,, Discrete and Contininuous Dynamical Systems, 26 (2010), 265.
[9]	A. Kolmogorov, I. Petrovski and N. Piscounov, Etude de l'équation de la diffusion avec croissance de la quantité de matiére et son application à un probléme biologique,, Bull. Univ. Moskou Ser. Internat. Sec. A, 1 (1937), 1.
[10]	P. 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. doi: 10.3934/dcdsb.2006.6.1175.
[11]	P. 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. doi: 10.1142/S0218202507002303.
[12]	F. Sánchez-Garduño, P. Maini and J. Pérez-Velázquez, A non-linear degenerate equation for direct aggregation and traveling wave dynamics,, Discrete and Continuous Dynamical Systems Series B, 13 (2010), 455.
[13]	F. Sánchez-Garduño and P. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations,, Journal of Mathematical Biology, 33 (1994), 163. doi: 10.1007/BF00160178.
[14]	L. Malaguti and C. Marcelli, Travelling wavefronts in reaction-diffusion equations with convection effects and non-regular terms,, Math. Nachr., 242 (2002), 148. 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,, Journal of Differential Equations, 195 (2003), 471. doi: 10.1016/j.jde.2003.06.005.
[16]	P. Pang, Y. Wang and J. Yin, Periodic solutions for a class od reaction-diffusion equations with $p$-Laplacian,, Nonlinear Analysis: Real World Applications, 11 (2010), 323. doi: 10.1016/j.nonrwa.2008.11.006.

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