= Solution
Let $F(T)$ denote the right-hand side of the temperature <partial differential equation>. For a periodic variation $\eta$, <integration by parts> gives
$$
\begin{aligned}
DV[T]\eta&=\left\langle\mu\nabla T\cdot\nabla\eta-\Delta T\,\Delta\eta-|\nabla T|^2\nabla T\cdot\nabla\eta\right\rangle\\
&=\left\langle\left[-\mu\Delta T-\Delta^2T+\nabla\cdot(|\nabla T|^2\nabla T)\right]\eta\right\rangle
=\langle F(T)\eta\rangle.
\end{aligned}
$$
Along the evolution, $T_t=F(T)$, so this is a <gradient flow> with increasing functional:
$$
\boxed{\frac{dV}{dt}=\langle T_t^2\rangle\geq0.}
$$
For $s=|\nabla T|^2$, completing the square gives
$$
\boxed{V[T]=\frac{\mu^2}{4}-\left\langle\frac{(s-\mu)^2}{4}+\frac{(\Delta T)^2}{2}\right\rangle\leq\frac{\mu^2}{4}.}
$$
Thus the <increasing-gradient functional for poorly conducting convection> converges to a finite limit along any globally smooth solution, and
$$
\int_0^\infty\langle T_t^2\rangle\,dt=V_\infty-V[T(0)]<\infty.
$$
In particular, there are arbitrarily late times at which $\|T_t\|_{L^2}$ is arbitrarily small. A nonstationary time-periodic solution is impossible: it would have a strictly positive increase of $V$ over a temporal period. More generally, if the orbit is precompact in a topology strong enough for the equation, its limiting states are stationary. With the conserved mean fixed, the displayed square completion bounds $\|\Delta T\|_{L^2}$, and periodic elliptic estimates give an $H^2$ bound; the usual smoothing and compactness for a globally regular parabolic solution support this limiting-state conclusion. The inequalities alone do not prove convergence to one uniquely specified equilibrium, and stationary families related by <translation symmetry> can remain.
Now take a smooth, nonconstant periodic roll $T_0(x)$. Multiplying its stationary <partial differential equation> by $T_0$ and using periodic <integration by parts> yields
$$
\boxed{\mu\langle T_0'^2\rangle-\langle T_0''^2\rangle-\langle T_0'^4\rangle=0.}
$$
The primes here are essential: the first and fourth terms involve $T_0'$, not $T_0$. Write $a=T_0'$, $b=T_0''$, $M_2=\langle a^2\rangle$, $M_4=\langle a^4\rangle$, and $N_2=\langle b^2\rangle$. The orthogonal-roll perturbation has
$$
|\nabla(T_0(x)+\delta T_0(y))|^2=a(x)^2+\delta^2a(y)^2,\qquad
\Delta(T_0(x)+\delta T_0(y))=b(x)+\delta b(y).
$$
The cross term in the squared <Laplacian> averages to zero since $\langle b\rangle=0$. Independence of the $x,y$ cell integrals then gives the exact <polynomial> difference
$$
\begin{aligned}
V[T_0(x)+\delta T_0(y)]-V[T_0(x)]
&=\frac{\delta^2}{2}(\mu M_2-N_2-M_2^2)-\frac{\delta^4}{4}M_4\\
&=\boxed{\frac{\delta^2}{2}(M_4-M_2^2)-\frac{\delta^4}{4}M_4}.
\end{aligned}
$$
Its leading <coefficient> is half the <variance> of $a^2$ over the period. That <variance> is strictly positive: a nonconstant smooth periodic function has a derivative which vanishes somewhere and is nonzero somewhere, so $a^2$ is not constant. Hence arbitrarily small transverse perturbations increase $V$ above the roll value.
One can turn this energy comparison into <linear instability>. The linearized evolution at the roll is the self-adjoint operator
$$
L\eta=-\mu\Delta\eta-\Delta^2\eta+\nabla\cdot\left(|\nabla T_0|^2\nabla\eta+2(\nabla T_0\cdot\nabla\eta)\nabla T_0\right).
$$
Choose the fixed-mean perturbation $\eta=T_0(y)-\langle T_0\rangle$. Since $D^2V[T_0](\eta,\eta)=\langle\eta L\eta\rangle=M_4-M_2^2>0$, the <Rayleigh quotient> is positive. The periodic self-adjoint fourth-order operator therefore has a positive <eigenvalue>. This proves the <transverse energy instability of nonconstant temperature rolls>: \b[every nonconstant roll is unstable wherever it exists].
The existence qualification matters. Constants are also stationary and are stable on the fixed-mean subspace for $0<\mu<1$. Moreover, the periodic <Poincaré inequality> gives $N_2\geq M_2$, so the roll identity implies $\mu M_2=N_2+M_4>M_2$ for a nonconstant $2\pi$-periodic roll. Such rolls require $\mu>1$ on this specified cell; the conclusion for all positive $\mu$ is conditional on a nonconstant roll being available.
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