Soluciones positivas y soluciones con frontera libre para ecuaciones singulares
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Universidad Industrial de Santander
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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.
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.
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.
Keywords
Singular equations, free boundary, Ecuaciones singulares, frontera libre