Hi I wonder someone could help me check my understanding of getting an inequality of the estimate for the 3rd term $\|u'_m\|_{L^2(0,T;H^{-1}(U))}$ correctly. This inequality need to be checked is the following $$|\langle u'_m,v\rangle|\le C(\|f\|_{L^2(U)}+\|u_m\|_{H_0^1(U)})\qquad (1)$$ The following are my steps to get to the claim above. First, Equation (16) leads to $$(u'_m,v^1)+B[u_m,v^1;t]=(f,v^1)$$ Then $$ \langle u'_m,v\rangle=(u'_m,v)=(u'_m,v^1)=(f,v^1)-B[u_m, v';t]$$ Now, similar treatment as the previous term, minikowski inequality leads to $$|(f,v')|\le \frac{1}{2}\|f\|_{L^2(U)}+\frac{1}{2}\|v'\|_{L^2(U)}\le\frac{1}{2}(\|f\|_{L^2(U)}+1)$$ since $\|v'\|_{L^2(U)}\le \|v'\|_{H_0^1(U)}\le 1$. On the other hand using the energy estimates for the bilinear functional, it follows that $$ B[u_m, v';t]\le \alpha\|u_m(t)\|_{H_0^1(U)}\|v^1(t)\|_{H_0^1(U)}\le \alpha\|u_m(t)\|_{H_0^1(U)}$$ Hence (1) follows. Any corrections?
Please help anybody. I really need to know explicitly what is going on here.