Advanced Search
Article Contents
Article Contents

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

Abstract / Introduction Related Papers Cited by
  • 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 $p=2$.
    Mathematics Subject Classification: Primary: 34B18, 34C37, 35K57.


    \begin{equation} \\ \end{equation}
  • [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), 11-30.


    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.


    H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Annales de l'Institut Henri Poincare- Analyse non lineaire, 9 (1992), 497-572.


    D. Bonheure and L. Sanchez, "Heteroclinic Orbits For Some Classes of Second and Fourth Order Differential Equations,'' Handbook of Differential Equations: Ordinary Differential Equations, Elsevier, 3 2006.doi: 10.1016/S1874-5725(06)80006-4.


    B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction,'' Progress in Nonlinear Differential Equations and their Applications, 60. Birkhauser Verlag, Basel, 2004.


    A. Hamydy, Travelling wave for absorption-convection-diffusion equations, Electronic Journal of Diff. Eq., 2006 (2006), 1-9.


    A. Sánchez-Valdés and B. Hernández-Bermejo, New travelling wave solutions for the Fisher-KPP equation with general exponents, Appl. Math. Lett., 18 (2005), 1281-1285.doi: 10.1016/j.aml.2005.02.016.


    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), 265-290.


    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), 1-25.


    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), 1175-1189.doi: 10.3934/dcdsb.2006.6.1175.


    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), 1351-1368.doi: 10.1142/S0218202507002303.


    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), 455-487.


    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), 163-192.doi: 10.1007/BF00160178.


    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.


    L. Malaguti and C. Marcelli, Sharp Profiles in degenerate and doubly degenerate Fisher-KPP equations, Journal of Differential Equations, 195 (2003), 471-496.doi: 10.1016/j.jde.2003.06.005.


    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), 323-331.doi: 10.1016/j.nonrwa.2008.11.006.

  • 加载中

Article Metrics

HTML views() PDF downloads(173) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint