int_0^T int_Omega u^m _tu_tdxdt + int_0^T int_Omega - Delta _p ^su^mu^m_tdxdt = int_0^T int_Omega fuu^m_t dxdt继续化简

首先，利用乘积法则将第一项中的 $(u^m)t u_t$ 展开，得到 \begin{align*} &\int_0^T {\int\Omega {({u^m}} } {)t}{u_t}dxdt\ =&\int_0^T {\int\Omega {u_t^m}{u_t}dxdt} + \int_0^T {\int_\Omega {mu_t^{m-1}} } ({u_t})^2dxdt\ =&\frac{1}{2}\int_0^T {\int_\Omega {(u_t^m)^2} dxdt} + \frac{1}{2}\int_0^T {\int_\Omega {(mu_t^{m-1})^2}} dxdt\ &\quad + \frac{1}{2}\int_0^T {\int_\Omega {u_t^{2m}}dxdt} - \frac{1}{2}\int_0^T {\int_\Omega {u_t^{2m-2}}}(u_t)^2dxdt. \end{align*} 接下来，利用分部积分公式将第二项中的 $(u^m)t$ 移到 $u^m$ 上方，得到 \begin{align*} &\int_0^T {\int\Omega {( - {\Delta p})} } ^s{u^m}{({u^m})t}dxdt\ =&\int_0^T {\int\Omega {(-\Delta_p)^s u^m} } ({u^m})t dxdt\ =&\int_0^T {\int\Omega {(-\Delta_p)^s u^m} } d(u^m) - \int_0^T {\int\Omega {(-\Delta_p)^s u^m}} u_t^m dxdt\ =&\int_0^T {\int_\Omega {(-\Delta_p)^{s+1} u^m}} dxdt - \int_0^T {\int_\Omega {(-\Delta_p)^s u^m}} u_t^m dxdt. \end{align*} 将上述两项代入原式，得到 \begin{align*} &\frac{1}{2}\int_0^T {\int_\Omega {(u_t^m)^2} dxdt} + \frac{1}{2}\int_0^T {\int_\Omega {(mu_t^{m-1})^2}} dxdt\ &\quad + \frac{1}{2}\int_0^T {\int_\Omega {u_t^{2m}}dxdt} - \frac{1}{2}\int_0^T {\int_\Omega {u_t^{2m-2}}}(u_t)^2dxdt\ &\quad + \int_0^T {\int_\Omega {(-\Delta_p)^{s+1} u^m}} dxdt - \int_0^T {\int_\Omega {(-\Delta_p)^s u^m}} u_t^m dxdt\ =&\int_0^T {\int_\Omega {f(u)({u^m}_t)} } dxdt. \end{align*