Above the threshold,

c^{2} k_{j}^{2} > 1/ (4 τ^{2})

, the square root in part (d) is real. Both fields oscillate, but the first has envelope

e^{- t / (2 τ)}

and the second

e^{+ t / (2 τ)}

. Thus the first is damped and the second amplified.

Below the threshold, set

σ_{j} = 1/ (4 τ^{2}) - c^{2} k_{j}^{2}

. The temporal factors are

ϕ_{1} : e^{[- 1/ (2 τ) \pm σ_{1}] t}, ϕ_{2} : e^{[+ 1/ (2 τ) \pm σ_{2}] t} .

(1)

They are nonoscillatory. For nonzero

k_{j}

0 < σ_{j} < 1/ (2 τ)

, 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 / τ}

for

ϕ_{1}

e^{t / τ}

for

ϕ_{2}

At equality, the repeated temporal root gives

(A + Bt) e^{- t / (2 τ)}

for the first field and

(C + D t) e^{t / (2 τ)}

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.