Take the $\widetilde H$ inner product of the finite-dimensional equation with $u_m$. Part a gives exact cancellation of the nonlinear term:
$$
\nu\|u_m\|^2=(f,u_m).
$$
By the <Cauchy-Schwarz inequality> and the Poincare inequality,
$$
(f,u_m)\leq|f|\,|u_m|
\leq\mu_1^{-1/2}|f|\,\|u_m\|.
$$
Thus every solution satisfies
$$
\boxed{
\|u_m\|\leq R,
\qquad
R=\frac{|f|}{\nu\mu_1^{1/2}}}.
$$
The bound is independent of $m$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 2 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-359/2/b/i/solution