= Solution
Multiply the <viscous scalar conservation law> by $-u_{xx}$, rather than estimating $F''$ after differentiating. <Integration by parts> yields
$$
\frac12\frac{d}{dt}\|u_x\|_2^2+\varepsilon\|u_{xx}\|_2^2
=\int F'(u)u_xu_{xx}
\leq M\|u_x\|_2\|u_{xx}\|_2
\leq\frac\varepsilon2\|u_{xx}\|_2^2+\frac{M^2}{2\varepsilon}\|u_x\|_2^2.
$$
Thus $d\|u_x\|_2^2/dt\leq(M^2/\varepsilon)\|u_x\|_2^2$. Applying the <Gronwall inequality> gives
$$
\boxed{\|u_x(t)\|_2^2\leq e^{M^2t/\varepsilon}\|u_x(0)\|_2^2,\qquad C_1=M^2/\varepsilon.}
$$
Only the assumed bound on $F'$ is used; a global bound on $F''$ is not needed for this <energy estimate>.
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