Heat Equation: Core and Solutions

This lecture covers the heat equation, discussing the core concepts such as d'Alembert solutions, boundary conditions, and the principle of Duhamel. It explores the extension of solutions over bounded domains and the dependence on initial values. The instructor presents the derivation of solutions and the characteristics of heat propagation. The lecture emphasizes the importance of understanding the diffusion coefficient and the behavior of solutions over time. Various examples and remarks are provided to illustrate the application of the heat equation in different scenarios.