A nonlinear degenerate Kolmogorov equation with singular initial data

This thesis focuses on the study of the heat equation with power-type nonlinearity and singular initial data. The main goal is to extend the results established in this classical framework to the broader setting of Kolmogorov operators with constant coefficients. The motivation behind this work is to understand how analytical techniques developed for the classical heat equation can be generalized to degenerate parabolic operators. The work is organized into two main parts. In the first part, we address the problem of local well-posedness for a semilinear heat equation with singular initial data in Lebesgue spaces. After recalling the main classical results by Fujita, Brezis, Cazenave and Weissler, and after reformulating the problem in the whole space, we develop an approach based on some L^p-L^q estimates for the heat semigroup, which makes it possible to apply the contraction principle in a suitably chosen Banach space. In this way, we establish the existence and uniqueness of local solutions, together with criteria for continuous dependence on the initial data, in the subcritical case. The second part of the study is devoted to Kolmogorov operators with constant coefficients, which include degenerate cases of great interest in kinetic theory and mathematical finance (for instance, generalized Fokker–Planck equation and Black&Scholes model). After recalling the notions of hypoellipticity and invariance under dilations, we study the fundamental solution associated with such operators, which plays a role analogous to the Gaussian heat kernel in the classical case. Moreover, we derive L^p−L^q estimates for the convolution with the fundamental solution, which constitute the central tool in handling the nonlinearity. Afterwards, we consider a Cauchy problem where the operator is a Kolmogorov operator belonging to the studied class. The analysis follows the strategy already developed in the heat case: thanks to the L^q-L^q estimates, it is possible to control the nonlinear term and reformulate the problem in an integral form. By applying the contraction mapping principle, we prove the existence and uniqueness of local mild solutions, even in the presence of singular initial data. In conclusion, we present some theorems which guarantee that the mild solution is also a classical solution. The entire analysis is conducted in the subcritical case. Thus, the thesis contributes to the understanding of the local behavior of solutions for a class of degenerate PDE, with possible applications both in theory and in modeling. Finally, a possible direction for future research would be to study the problem in a bounded domain Ω, or to extend the analysis to the critical and supercritical cases.