 Solution

Above the threshold, $c^2{k}_j^2>1/(4{\tau }^2)$, the square root in part (d) is real. Both fields oscillate, but the first has envelope $\boxed{e^{-t/(2\tau )}}$ and the second $\boxed{e^{+t/(2\tau )}}$. Thus the first is damped and the second amplified.

Below the threshold, set ${\sigma }_j=\sqrt{1/(4{\tau }^2)-c^2{k}_j^2}$. The temporal factors areThey are nonoscillatory. For nonzero $k_j$, $0<{\sigma }_j<1/(2\tau )$, so both first-field branches decay and both second-field branches grow. At $k_j=0$, one branch is constant; the other is $e^{-t/\tau }$ for ${\varphi }_1$ or $e^{t/\tau }$ for ${\varphi }_2$.

At equality, the repeated temporal root gives $\boxed{(A+Bt)e^{-t/(2\tau )}}$ for the first field and $\boxed{(C+Dt)e^{t/(2\tau )}}$ for the second. The full two-field system is generically unstable on both sides because of the amplified sector. This is compatible with its autonomous Lagrangian: the total conserved energy is an indefinite quadratic form, rather than a positive energy forcing every component to remain bounded.