Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 105 / 4 / a

In three dimensions, the Sobolev inequality gives $\Vert v{\Vert }_{L^6(U)}\le C_U\Vert v{\Vert }_{H_0^1(U)}$. Thereforeand hence

Using $w^3-{\tilde{w}}^3=(w-\tilde{w})(w^2+w\tilde{w}+{\tilde{w}}^2)$, the Holder inequality and the same Sobolev embedding give at each timeTaking the $L^2$ norm in time supplies the required estimate with the factor $T^{1/2}$.