Bounding an integral (Sobolev spaces and duality pairing), explanation needed

Question

I don't understand how the inequality is derived here:

I understand the equality but not how he gets the last line. I get a triple integral and I don't know to deal with that.

Answer 1

I'll omit the subscript $\delta$. It seems that the variables of integration on the second line should be $t$ and $\tau$. By the definition of dual norm in $W^{-1,p'}$, for every fixed $\tau,t$ $$\langle b(u)_t,u(\tau+h)-u(\tau)\rangle \le \|b(u)_t\|_{W^{-1,p'}} \|u(\tau+h)-u(\tau)\|_{W_0^{1,p}} \\ \le \|b(u)_t\|_{W^{-1,p'}}\big(\|u(\tau+h)\|_{W_0^{1,p}} +\| u(\tau)\|_{W_0^{1,p}}\big) $$ The second factor does not depend on $t$ (it seems), so when we integrate over $t$, Hölder's inequality yields $$\int_\tau^{\tau+h} \|b(u)_t\|_{W^{-1,p'}} \cdot 1 \,dt \le \|b(u)_t\|_{L^{p'}(0,T;W^{-1,p'})} \, h^{1/p}$$ times the second factor, the integral of $\big(\|u(\tau+h)\|_{W_0^{1,p}} +\| u(\tau)\|_{W_0^{1,p}}\big) $.