 Solution

The diffusion equation is linear, so superposition givesEach term separately satisfies the equation, and their sum recovers the stated initial data at $t=0$. Its e-folding times are ${\tau }_i={\lambda }_i^2/(4{\pi }^2\kappa )$, so ${\lambda }_1<{\lambda }_2$ implies ${\tau }_1<{\tau }_2$. The shorter-wavelength oscillation loses a given fraction of its amplitude first. It approaches zero exponentially rather than disappearing at a finite instant. This preferential damping explains smoothing of fine-scale temperature structure.