Given the parabolic PDE : $$u_t-\triangle u =|u|^{p-1}u, \,\,\,\, x \mbox{ in }\Omega, \,\, t>0$$
$$u=0 \,\,\mbox{ in } \,\,\partial \Omega,\,\,t>0$$ $$u(x,0) = u_0(x), \,\,\,\, x \mbox{ in }\Omega$$ its corresponding energy functional is $$ E(u)=\int_\Omega\Bigl(\frac12|\nabla u|^2-\frac1{p+1}|u|^{p+1})dx. $$ I'm trying to get the derivative of this functional energy, to verify that it's nonincreasing. I've already been able to show that $\dfrac{d}{dt}\int_\Omega \frac12|\nabla u|^2 dx= -2\int_\Omega u_t(t)\Delta u(t)dx$, but I was in doubt if I did it right in the calculation that $ -\frac1{p+1} \dfrac{d}{dt}\int_\Omega|u|^{p+1}dx = -\int_\Omega |u|^{p-1}uu_t dx$. I thought this way, to derive the term $|u|^{p+1}$ use the chain rule, then $$-\frac1{p+1} \dfrac{d}{dt}\int_\Omega|u|^{p+1}dx = -\frac1{p+1} \int_\Omega \dfrac{d}{dt}|u|^{p+1} dx = - \int_\Omega |u|^{p} \dfrac{d}{dt}|u| dx = - \int_\Omega |u|^{p} \left( \sum_i \dfrac{u_i}{|u|}(u_i)_t \right) dx = -\int_\Omega |u|^{p-1}uu_t dx.$$
Can you confirm this calculation is correct?