Discussion
Let \(u = u(x, t)\) be the solution of
\(\frac{\partial u}{\partial t} - 4\frac{\partial^{2} u}{\partial x^{2}} = 0,\) \(0 \lt x \lt 1,\ t \gt 0,\)
\(u(0, t) = u(1, t) = 0,\ t \ge 0,\)
\(u(x, 0) = \sin(\pi x),\ 0 \le x \le 1.\)
Define \(g(t) = \int_{0}^{1}(u(x, t))^{2}dx\), for \(t \gt 0\). Which one of the following is correct?
Answer: A
\(g\) is decreasing on \((0, \infty)\) and \(\lim_{t \to \infty} g(t) = 0\).
The initial data is a single Fourier sine mode, so the solution is \(u(x, t) = e^{-4\pi^{2}t}\sin(\pi x)\). Then
\[ \begin{aligned} g(t) &= e^{-8\pi^{2}t}\int_{0}^{1}\sin^{2}(\pi x)\,dx \\ &= \tfrac{1}{2}e^{-8\pi^{2}t} \end{aligned} \]
This is strictly decreasing and tends to 0. The heat equation with zero boundary values always dissipates energy, so the increasing options are wrong, and 1/4 would need the solution to settle to a non-zero state, which zero boundary conditions forbid.
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