Soluciones positivas y soluciones con frontera libre para ecuaciones singulares

• Autores: Juan Dávila, Marcelo Montenegro
• Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 28, Nº. 2, 2010 (Ejemplar dedicado a: Revista Integración), págs. 85-100
• Idioma: español
• Títulos paralelos:
  • Remarks on positive and free boundary solutions to a singular equation
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    La ecuación −∆u = χ{u>0} (− 1/(u^β) + λf(x, u) en ∂Ω con condición de frontera de tipo Dirichlet en ∂Ω posee una solución uλ ≥ 0 para λ > 0. Si λ es menor que una constante λ ∗ la solución es nula dentro de una región del dominio, y para λ > λ∗ la solución es positiva y estable. Obtenemos la regularidad óptima de uλ aun con la frontera libre. Si 0 < λ < λ∗ las soluciones de la ecuación parabólica singular ut − ∆u + 1/(u^β) = λf(u) son nulas en tiempo finito, y para λ > λ∗ las soluciones son positivas y globalmente definidas. Palabras claves: Ecuaciones singulares, frontera libre.

  • English

    The equation −∆u = χ{u>0} (− 1/(u^β) + λf(x, u) in Ω with Dirichlet boundary condition on ∂Ω has a maximal solution uλ ≥ 0 for every λ > 0. For λ less than a constant λ ∗ the solution vanishes inside the domain, and for λ > λ∗ the solution is positive and stable. We obtain optimal regularity of uλ even in the presence of the free boundary. If 0 < λ < λ∗ the solutions of the singular parabolic equation ut − ∆u + 1/(u^β) = λf(u) quench in finite time, and for λ > λ∗ the solutions are globally positively defined.

     

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