= Solution
The nonstationary part of $u$ decays exponentially, whereas the integrating factor grows quadratically. For fixed $x>0$,
$$
\frac{v(x,t)}{3xt^2/\pi}
=\left(1+\frac1t\right)^2\left(1+\frac{e^{-4t}\sin2x}{3x}\right)\longrightarrow1.
$$
Consequently
$$
\boxed{v(x,t)\sim\frac{3xt^2}{\pi}\qquad(t\to\infty).}
$$
At $x=0$, both expressions are identically zero; the ratio formulation is understood for $x>0$. The diffusion modes have died away, leaving the lifted linear profile multiplied by the reaction-induced amplitude factor.
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