Heat Equation: Properties of Solutions
Summary
We continue the treatment of the heat equation and cover properties of the solutions of the initial-boundary-value problem (IBVP) $$ \left\{ \begin{alignedat}{2} u_t -\Delta u &= 0\quad &&\text{in}~U_T\\ u&=g\quad&&\text{on}~\Gamma_T \end{alignedat} \right. $$ where $U_T:=U \times (0,T]$ is called a parabolic cylinder. Here $U\subset \mathbb{R}^n$ is open and bounded, $T>0$ is a some terminal time, and $\Gamma_T:=\overline{U}_T \setminus U_T$ is the parabolic boundary of $U_T$.
We cover properties of the solutions of (IBVP), including
- Mean value property
- Maximum principles & uniqueness
- Regularity
We then consider the initial-value problem posed on $\mathbb{R}^n\times (0,T)$, discuss the maximum principle and uniqueness.
Full Set of Lecture Notes
The notes for this lecture are available here (17 pages).