I want to ask about the articl: http://ejde.math.txstate.edu/Volumes/2011/18/chen.pdf

In the lamma4.4 and 4.5 and 4.6 the required convergence is strong but in the proof she indicate just the weak convergence so why?

what does it mean $L^q(0,T;\ H^{-r}(\Omega))$ or how can I read it mathematicaly?