Heat Equation
Jan 1, 2021
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1 min read
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.
