Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 13A / c

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- Solution c

Solution

Each spatial mode acquires the factor

exp (\int_{0}^{t} μ (s) d s - D k_{n}^{2} t) = exp ((μ_{0} - D k_{n}^{2}) t + μ_{1} sin t) .

The oscillatory factor

e^{μ_{1} s i n t}

is bounded and does not affect the long-term exponential rate. With two open ends, growth therefore occurs exactly when

μ_{0} > \frac{D π ^{2}}{L ^{2}} .

Only the time average of this spatially uniform fluctuating growth rate controls the asymptotic threshold.

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