= Solution
Define the squared <L2 norm> $E(t)=\int_0^1u(x,t)^2\,dx$. Multiplication of the <semilinear partial differential equation> by $u$, followed by <integration by parts>, gives
$$
\frac12E'(t)=[u u_x]_0^1-\int_0^1u_x^2\,dx
+a(t)\int_0^1u\sin^n u\,dx.
$$
At $x=1$ the <Dirichlet boundary condition> makes the boundary term zero. At $x=0$ the <Robin boundary condition> is $u_x(0,t)=\alpha u(0,t)$, so subtraction of the lower endpoint gives $-\alpha u(0,t)^2$. In particular the sign is dissipative. Thus the exact <energy identity> is
$$
\frac12E'(t)+\int_0^1u_x^2\,dx+\alpha u(0,t)^2
=a(t)\int_0^1u\sin^n u\,dx,
$$
which implies
$$
\boxed{\frac12\frac d{dt}\int_0^1|u|^2\,dx
\leq a(t)\int_0^1u\sin^n u\,dx.}
$$
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