Show that $L$ it is a continuous operator.

Question

Let $L(.)$ a linear operator in $W$ so that

$$L(\phi)=\int_0^T<f,\phi>_{H^{-1}, \ H_0^1}dt+(u_0,\phi(0))_{L^2}.$$

Then $L(.)$ it is continuous in $W$. Where $f \in L^2(0,T; H^{-1}(\Omega))$ and $u_0 \in L^2(0, T; H_0^1(\Omega))$

$$W=\{\phi \in L^2(0,T;H_0^1(\Omega));\ \phi' \in L^2(0,T;L^2(\Omega)) \ \mbox{and} \ \phi(T)=0\}$$

and

$$||\phi||^2_W=\int_0^T||\phi(t)||_{H_0^{1}}^2dt +\int_0^T|\phi'(t)|_{L^2}^2dt.$$

Answer 1

Note that $W^{1,1}(0,T; L^2(\Omega)) \hookrightarrow C([0,T]; L^2(\Omega))$. Hence you can bound $\|\phi(0)\|_{L^2(\Omega)} \le C \, \|\phi\|_{H^1(0,T; L^2(\Omega))}$.