[再寄小读者之数学篇](2014-06-19 两个分部积分)

For $2<q<\infty$,

$$\beex \bea -\int \lap \bbu \cdot |\bbu|^{q-2}\bbu &=\int \p_iu_j \p_i\sex{|\bbu|^{q-2}u_j}\\ &=\int \p_iu_j \p_i|\bbu|^{q-2}u_j+\int \p_iu_j|\bbu|^{q-2}\p_iu_j\\ &=\cfrac{1}{2}\int \p_i|\bbu|^2\cdot \p_i|\bbu|^{q-2} +\int |\bbu|^{q-2}|\n\bbu|^2\\ &=\cfrac{q-2}{2}\int |\bbu|\p_i|\bbu|\cdot |\bbu|^{q-3}\p_i|\bbu| +\int |\bbu|^{q-2}|\n\bbu|^2\\ &=\cfrac{q-2}{2}\int |\bbu|^{q-2}|\n|\bbu||^2 +\int|\bbu|^{q-2}|\n\bbu|^2\\ &=\cfrac{2(q-2)}{q^2}\int ||\bbu|^{\frac{q}{2}-1}|^2 +\int |\bbu|^{q-2}|\n\bbu|^2;\\ \cfrac{\rd}{\rd t}|\bbu|^q &=\cfrac{\rd}{\rd t}(|\bbu|^2)^\frac{q}{2}\\ &=\cfrac{q}{2}(|\bbu|^2)^{\frac{q}{2}-1}\cdot 2\bbu\cfrac{\rd \bbu}{\rd t}\\ &=q|\bbu|^{q-2} \bbu \cdot \cfrac{\rd \bbu}{\rd t}. \eea \eeex$$