Part CCSIR NET December 2024the-energy-identity-comes-from-one-integration-by-parts

The energy identity comes from one integration by parts

Let u = u(x, t) be the solution of the initial-boundary value problem $\partial u / \partial t = \partial^{2} u / \partial x^{2}$ on $(0, 1) \times (0, \infty)$ , with u(x, 0) = 4x(1 − x) for x ∈ [0, 1] and u(0, t) = u(1, t) = 0 for t ≥ 0. Then which of the following statements are true?

A. $lim (t \to \infty) u (x, t) = 0$ for all x ∈ (0, 1)✓
B.u(x, t) = u(1 − x, t) for all x ∈ (0, 1), t > 0✓
C. $\int_{0}^{1} (u (x, t))^{2}$ dx is a non-increasing function of t✓
D. $\int_{0}^{1} (u (x, t))^{2}$ dx is a non-decreasing function of t

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: All the classical equations satisfy a maximum principle

The energy identity comes from one integration by parts

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