Nonlinear and nonlocal diffusion equations. Qualitative theory and asymptotic behaviour

• Autores: Alessandro Audrito
• Directores de la Tesis: Susanna Terracini (dir. tes.) , Juan Luis Vázquez (dir. tes.)
• Lectura: En la Universidad Autónoma de Madrid ( España ) en 2018
• Idioma: inglés
• Número de páginas: 229
• Tribunal Calificador de la Tesis: Enrico Priola (presid.) , Matteo Bonforte (secret.) , Veronica Felli (voc.) , Juan Luis Vázquez (voc.)
• Enlaces
  • Tesis en acceso abierto en: Biblos-e Archivo
• Resumen

  • The thesis is divided in two main parts. We report here a short resume of both of them.

    In the first part, we study a class of reaction equations with nonlinear diffusion. The famous Fisher-KPP reaction-diffusion model combines linear diffusion with the typical KPP reaction term, and appears in a number of relevant applications in biology and chemistry. It is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions of the problem posed in the real line. The existence of propagation waves with finite speed has been confirmed in some related models and disproved in others.

    - We first investigate the corresponding theory when the linear diffusion is replaced by the slow doubly nonlinear diffusion and we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions. A similar study is performed in the critical case that we call pseudo-linear, i.e., when the operator is still nonlinear but has homogeneity one. With respect to the classical model and the pseudo-linear case, the slow travelling waves exhibit free boundaries.

    - We secondly investigate the corresponding theory with fast doubly nonlinear diffusion and we find that general solutions show a non-TW asymptotic behaviour, and exponential propagation in space for large times. Finally, we prove precise bounds for the level sets of general solutions of the initial value problem posed in any spacial dimension. In particular, we show that location of the level sets is approximately linear for large times, when we take spatial logarithmic scale, finding a strong departure from the linear case, in which appears the famous Bramson logarithmic correction.

    - We finally investigate the existence of waves with constant propagation speed, when the linear diffusion is replaced by the slow doubly nonlinear diffusion and reaction term is of bistable type. This framework presents interesting deviances from the Fisher-KPP one like, for instance, the apparition of a threshold phenomena. We find a different family of travelling waves that are employed to describe the wave propagation of more general solutions and to study the stability/instability of the steady states. A similar study is performed in the pseudo-linear case.

    In the second part, we investigate the nodal properties of solutions to a nonlocal Heat Equation. We characterise the possible blow-ups and we examine the structure of the nodal set of such solutions. More precisely, we prove that their nodal set has at least parabolic Hausdorff codimension one in the space-time, and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to a Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results like the Federer Reduction Principle and the Parabolic Whitney Extension.