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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 ∂u/∂t=∂2u/∂x2\partial{}u/\partial{}t = \partial^{2}u/\partial{}x^{2} on (0,1)×(0,∞)(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?

  1. A.lim⁡(t→∞)u(x,t)=0\lim(t\to\infty) u(x, t) = 0 for all x ∈ (0, 1)✓ Correct
  2. B.u(x, t) = u(1 − x, t) for all x ∈ (0, 1), t > 0✓ Correct
  3. C.∫01(u(x,t))2\int_{0}^{1}(u(x, t))^{2}dx is a non-increasing function of t✓ Correct
  4. D.∫01(u(x,t))2\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, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: All the classical equations satisfy a maximum principle

More on this topic

The chapter behind this: The three classical equations — free to read

From Partial Differential Equations › Laplace, heat and wave equations: separation of variables

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