Summary of Separation of Variables — Question 8

Question 8 – Solution

Strategy. A separated coefficient equation can be singular or unstable; stationarity, uniqueness and attraction are separate questions.

Step 1: Test the stationary kernel. For $\mathcal L=-d^2/dx^2-4$, the eigenvalues on $\sin(nx)$ are $n^2-4$. The kernel is spanned by $\sin(2x)$. Integration by parts, using both zero endpoints, gives $0=\int_0^\pi(\mathcal Lu)\sin(2x)\,dx=\int_0^\pi f\sin(2x)\,dx=a\pi/2$. Thus $\boxed{a=0}$ is necessary.

Step 2: Solve the compatible stationary equations. When $a=0$, divide the first and third coefficients by $1-4=-3$ and $9-4=5$: $$\boxed{u=-\tfrac 13\sin x+\tfrac 15\sin(3x)+C\sin(2x),\qquad C\in\mathbb R.}$$ Direct substitution proves sufficiency. Orthogonality to the kernel forces $C=0$, and without it the stationary field is not unique.

Step 3: Evolve from rest. The modal equations are $q_1'=3q_1+1$, $q_2'=a$, $q_3'=-5q_3+1$. Their zero-initial-value solutions give $$\boxed{v=\frac{e^{3t}-1}{3}\sin x+at\sin(2x) +\frac{1-e^{-5t}}5\sin(3x).}$$ Nonzero $a$ drives linear growth in the kernel mode. Separately, the first mode grows exponentially because $\mathcal L$ has eigenvalue $-3$. Therefore even compatible forcing does not make the stationary family an attractor from rest.

Step 4: Identify the bounded initial states. For the prescribed three-mode initial state, $$q_1=-\tfrac 13+(b_1+\tfrac 13)e^{3t},\qquad q_2=b_2+at,\qquad q_3=\tfrac 15+(b_3-\tfrac 15)e^{-5t}.$$ Orthogonality prevents cancellation between spatial modes. Boundedness holds exactly when $$\boxed{a=0,\qquad b_1=-1/3.}$$ The coefficients $b_2,b_3$ are unrestricted. Then the uniform limit is $-\tfrac 13\sin x+b_2\sin(2x)+\tfrac 15\sin(3x)$. Eliminating the unstable component and satisfying the kernel compatibility are both necessary.