Thesis defense

Abstract

In this thesis defence, we present four topics in Partial Differential Equations:

         We begin by developing a Widder-type theory for nonlocal heat equations involving certain Lévy operators. We establish uniqueness of nonnegative classical and very weak solutions with a given initial trace and the existence of an initial traces. We also prove a representation formula for nonnegative classical solutions. Here, we sumarize the results of [5].

         Next, we study the Cauchy problem on hyperbolic space for the heat equation with a Fisher-KPP type forcing term. Our goal is to understand how hyperbolic geometry affects the dynamics of solutions. We address the question of propagation versus extinction, including the critical case. In the case of propagation, we show that if the initial datum has a certain symmetry, the solution converges asymptotically to a traveling wave of minimal speed in a moving frame. The choice of this moving frame  depends on the symmetries of the initial datum, which, in turn, is closely related to the three types of isometries in hyperbolic space: elliptic, hyperbolic, and parabolic. [1].  

          Latter, we look for the convex hull of a set using the geometric evolution by minimal curvature of a hypersurface that surrounds the set. To find the convex hull,we study the large time behavior of solutions to an obstacle problem for the level set formulation of the geometric flow driven by the minimum of the principal curvatures. Using a game-theoretical approximation, we prove that the superlevel set of the solution converges to the convex hull of the obstacle as time goes to infinity. [4].

          Finally, we study an elliptic problem related to iterations of the obstacle problem for different operators. We work  on one hand in a Lipschitz domain with differential operators and on the other hand in  regular trees with mean-value-type operators. In both case, by solving the obstacle problem iteratively, we prove that the limit yields a solution to the two membranes problem corresponding to the operators involved. [2,3].

·      [1] M.M. González, I. Gonzálvez and F. Quirós. “Traveling wave behavior for Fisher- KPP equations on the hyperbolic space”. Coming soon on arXiv.

·      [2] I. Gonzálvez, A. Miranda and J.D Rossi. “Monotone iterations of two obstacle problem with different operators”  J. Elliptic Parabol Equations. 10.1 2024,

       link.springer.com/article/10.1007/s41808-024-00268-6 

·      [3] I. Gonzálvez, A. Miranda and J.D Rossi. “The two membranes problem in a regular tree”. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 2025

 resolve.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/abs/two-membranes-problem-in-a-regular-tree/C6C38C4D759A193C02FD4F49D5ED067A 

·      [4] I. Gonzálvez, A. Miranda, J.D Rossi and, J. Ruiz-Cases. “Finding the convex hull of a set using the flow by minimal curvature with an obstacle. A game theoretical approach” 2025. Accepted  in Communications in Analysis and Geometry. 

arxiv.org/abs/2409.06855 

·      [5]  I. Gonzálvez,  F. Quirós, F. Soria and Z. Vondraček. “On the nonlocal heat equation for certain Lévy operators and the uniqueness of positive solutions” 2025. arXiv:2504.04246

 arxiv.org/abs/2504.04246