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On the interfaces in a nonlocal quasilinear degenerate equation arising in population dynamics

  • Jesus Ildefonso Diaz
  • Toshitaka Nagai
  • Sergei I. Shmarev
Article

Abstract

We study regularity and propagation properties of interfaces separating regions where nonnegative weak solutions of the Cauchy problem for the equation
$$u_t = (u^m )xx + \left[ {u\left( {\int_{ - \infty }^x {u(y,t)dy} - \int_x^\infty {u(y,t)dy} } \right)} \right]_x , m > 1,$$
are strictly positive or equal to zero. It is shown that under suitable conditions on the initial data the interfaces are (not necessarily monotone)C -curves and they do not lose this regularity at their turning points. The study of the interface regularity is performed via Lagrangian coordinates. We show that the initial behaviorof interfaces is determined by the character of growth of the initial datum near the endpoints of the initial support. Estimates (from below and from above) on the width of the positivity set of solutions are also obtained.

Key words

interface nonlocal equation degenerate parabolic equation Lagrangian coordinate 

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Copyright information

© JJIAM Publishing Committee 1996

Authors and Affiliations

  • Jesus Ildefonso Diaz
    • 1
  • Toshitaka Nagai
    • 2
  • Sergei I. Shmarev
    • 3
    • 4
  1. 1.Departmento de Matemática aplicadaUniversidad Complutense de MadridMadridSPAIN
  2. 2.Department of MathematicsKyushu Institute of TechnologyKitakyushuJapan
  3. 3.Lavrentiev Institute of HydrodynamicsNovosibirskRussia
  4. 4.Departamento de Matemática Aplicada, Facultad de CienciasUniversidad de OviedoOviedoSpain

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