= Solution
Since $U\subset\mathbb R^3$ is bounded and smooth, the <Sobolev embedding theorem> gives continuous embeddings $H^1(U)\hookrightarrow L^4(U)$ and $H^1(U)\hookrightarrow L^6(U)$. Because $|\sin s|\leq1$,
$$
\|w(t)^2\sin w(t)\|_{L^2(U)}
\leq\|w(t)\|_{L^4(U)}^2
\leq C\|w(t)\|_{H^1(U)}^2.
$$
Taking the $L^2$ norm in time gives
$$
\boxed{\|w^2\sin w\|_{L^2(U_T)}
\leq\beta T^{1/2}\|w\|_{L_t^\infty H_x^1}^2}.
$$
For $F(s)=s^2\sin s$, the <mean value theorem> and $|F'(s)|\leq2|s|+s^2$ imply
$$
|F(r)-F(s)|
\leq C|r-s|\bigl(|r|+|s|+|r|^2+|s|^2\bigr).
$$
Apply the <Generalized Holder inequality> in space, using $L^4$ for the quadratic products and $L^6\cdot L^3$ for the cubic products, and then use the two Sobolev embeddings. Pointwise in time this yields
$$
\|F(w)-F(\widetilde w)\|_2
\leq C\|w-\widetilde w\|_{H^1}
\left(\|w\|_{H^1}+\|w\|_{H^1}^2
+\|\widetilde w\|_{H^1}+\|\widetilde w\|_{H^1}^2\right).
$$
Taking the $L^2$ norm in time proves the required estimate with the factor $\gamma T^{1/2}$.
Back to article page
