January  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

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

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 $p=2$.
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.   Google Scholar

[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.  Google Scholar

[3]

H. Berestycki and L. Nirenberg, Travelling fronts in cylinders,, Annales de l'Institut Henri Poincare- Analyse non lineaire, 9 (1992), 497.   Google Scholar

[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.  Google Scholar

[5]

B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction,'', Progress in Nonlinear Differential Equations and their Applications, (2004).   Google Scholar

[6]

A. Hamydy, Travelling wave for absorption-convection-diffusion equations,, Electronic Journal of Diff. Eq., 2006 (2006), 1.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[3]

H. Berestycki and L. Nirenberg, Travelling fronts in cylinders,, Annales de l'Institut Henri Poincare- Analyse non lineaire, 9 (1992), 497.   Google Scholar

[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.  Google Scholar

[5]

B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction,'', Progress in Nonlinear Differential Equations and their Applications, (2004).   Google Scholar

[6]

A. Hamydy, Travelling wave for absorption-convection-diffusion equations,, Electronic Journal of Diff. Eq., 2006 (2006), 1.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021008

[2]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[3]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047

[4]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[5]

Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control & Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030

[6]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[7]

Mia Jukić, Hermen Jan Hupkes. Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020402

[8]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[9]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159

[10]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387

[11]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[12]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

[13]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[14]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[15]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[16]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293

[17]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

[18]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[19]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[20]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (99)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]