Heat Equation
Summary
In this lecture we begin our treatment of the heat equation $$ u_t - \Delta u = 0, $$ where the unknown function is $u=u(x,t)$ with $u\colon \mathbb{R}^n\times(0,\infty)$. We construct the fundamental solution using dilation symmetry of the heat operator and then give a representation for solutions of the initial-value problem (IVP) $$ \left\{ \begin{alignedat}{2} u_t -\Delta u &= 0\quad &&\text{in}~\mathbb{R}^n \times(0,\infty)\\ u&=g\quad&&\text{on}~\mathbb{R}^n \times \{t=0\} \end{alignedat} \right. $$ that is, we show that $$ u(x,t):=(\Phi * g)(x,t) $$ satisfies (IVP) and that $u \in C^{\infty}\left(\mathbb{R}^{n} \times(0, \infty)\right)$ provided $g\in C\left(\mathbb{R}^{n}\right) \cap L^{\infty}\left(\mathbb{R}^{n}\right)$.
We then move on to the treatment of the inhomogeneous problem and cover Duhamel’s principle.