Summary of Separation of Variables — Question 9

Question 9 – Solution

Strategy. An eigenfunction expansion can become nearly one mode without ever being exactly one product at a finite time.

Step 1: Superpose the two decays. The eigenvalues are one and four, so $$\boxed{u=e^{-t}\sin x+e^{-4t}\sin(2x).}$$ The PDE and every datum follow term by term. Separation constructs building blocks; linearity allows their sums, whose distinct temporal factors need not combine into one product.

Step 2: Derive the restriction on a product. If $u=XT$ is a nonzero classical product, then $XT'=X''T$. Choose a time with $T\ne 0$ and points where $X\ne 0$. The ratios satisfy $T'/T=X''/X=-\lambda$, independent of both variables. The differential identity $X''=-\lambda X$ extends across spatial zeros by continuity; the temporal ODE $T'=-\lambda T$ extends likewise. Zero boundary values give the Dirichlet eigenproblem. The trivial zero product is separate. Division at zeros is unnecessary and cannot create extra eigenfunctions.

Step 3: Use a factorization invariant. Every product has determinant $$u(x_1,t_1)u(x_2,t_2)-u(x_1,t_2)u(x_2,t_1)=0.$$ For the stated points, the four entries are $$u(x_1,0)=\sqrt 3,\quad u(x_2,0)=0,\quad u(x_1,\log 2)=\frac{9\sqrt 3}{32},\quad u(x_2,\log 2)=\frac{7\sqrt 3}{32}.$$ Their determinant is $\boxed{21/32\ne 0}$, proving that the exact field is not one product.

Step 4: Quantify approximate separation. The normalized field is $e^tu=\sin x+e^{-3t}\sin(2x)$, so $$\boxed{\|e^tu-\sin x\|_\infty=e^{-3t},\qquad t\ge\tfrac 13\log(1/\varepsilon)\ \Longrightarrow\ \|e^tu-\sin x\|_\infty\le\varepsilon.}$$ Equality in the norm follows at a point where $|\sin(2x)|=1$. The plotted ratios correspond to $t=0,(\log 2)/3,(\log 10)/3$ and the limiting profile. Approximation by a surviving mode does not retroactively justify a single-product initial ansatz.