Nonlinear and nonlocal diffusion equations

• Autores: Diana Stan
• Directores de la Tesis: Juan Luis Vázquez (dir. tes.)
• Lectura: En la Universidad Autónoma de Madrid ( España ) en 2014
• Idioma: inglés
• Tribunal Calificador de la Tesis: Xavier Cabré Vilagut (presid.), Matteo Bonforte (secret.), Giuseppe Mingione (voc.), Jesús Ildefonso Díaz Díaz (voc.), Daniel Faraco Hurtado (voc.)
• Materias:
  • Matemáticas
    • Análisis y análisis funcional
      • Ecuaciones diferenciales en derivadas parciales
• Enlaces
  • Tesis en acceso abierto en: Biblos-e Archivo
• Resumen

  • We consider three different models of nonlinear diffusion equations. The prototype is the classical Porous Medium Equation u_t=\Delta u^m.

    The first model is the Doubly Nonlinear Diffusion Equation u_t=\Delta_p u^m for which we discuss the asymptotic behavior of the homogenous Dirichlet problem.

    The second model is The Fisher-KPP Equation with nonlinear fractional diffusion u_t+ (-\Delta)^s u^m=u(1-u) for x \in \mathbb{R}^N, t>0. We prove that the level sets of the solution propagate exponentially fast in time.

    The third model is the Porous Medium Equation with fractional pressure u_t=\nabla \cdot(u^{m-1}\nabla P), P=(-\Delta)^{-s}u for x\in \mathbb{R}^N, t>0. Our main result concerns the effect of the nonlinearity on the finite speed of propagation of the solutions.