When I read Zuazua's paper "Exponential decay for the semilinear wave equation with locally distributed damping" CPDE 1990, I had a problem with inequality estimation. That is (1.10)$\|\psi\|^{2}_{H^{1}(\Omega\times(0,T))}\leq C\|f(u)+a(x)u_{t}\|^{2}_{L^{1}(0,T;L^{2}(\Omega))}$.
The author says that by standard analysis we can get(1.10), and my attempt as below: \begin{equation}\begin{cases} \psi_{tt}-\Delta\psi=-f(u)-a(x)u_{t}, in \ \Omega\times(0,\infty),\\\psi=0,\ on\ \Gamma\times(0,\infty),\\ \psi(0)=\psi_{t}(0)=0,\ in\ \Omega \end{cases}\end{equation} Maybe multiply $\psi_{t}$ to the above equation: and integrate it in $[0,\tau]\times\Omega$ we have $\int_{\Omega}\psi_{t}^{2}(\tau,x)dx+\int_{\Omega}|\nabla\psi(\tau,x)|^{2}dx=\int_{0}^{\tau}\int_{\Omega}\psi_{t}(-f(u)-a(x)u_{t})dxdt$
Next, we integrate $\tau$ from $0$ to $T$, we have $\int_{0}^{T}\int_{\Omega}\psi_{t}^{2}(\tau,x)dxd\tau+\int_{0}^{T}\int_{\Omega}|\nabla\psi(\tau,x)|^{2}dxd\tau=\int_{0}^{T}\int_{0}^{\tau}\int_{\Omega}\psi_{t}(-f(u)-a(x)u_{t})dxdtd\tau$, then we use Cauchy inequality with $\epsilon$ to get the final (1.10) of that paper.
BUT next the author says that $C$ also depend on $\|f'\|_{L^{\infty}}$. Why not $\|f\|_{L^{\infty}}$? and the author writes (1.10) as $$\|\psi\|^{2}_{H^{1}(\Omega\times(0,T))}\leq C\|f(u)+a(x)u_{t}\|^{2}_{L^{1}(0,T;L^{2}(\Omega))}$$ why $L^{1}_{t}L^{2}_{x}$? Why not $L^{2}_{t}L^{2}_{x}$?
In addition, in page 213 of that paper the author also says that "We have implicitly used the fact that $|f(s)|\leq C|s|$ and $|F(s)|\leq C|s|^{2}$" Where do we use the fact $|F(s)|\leq C|s|^{2}$? in which $F(s)=\int_{0}^{s}f(x)dx$ Is it for us to take advantage of $\psi=u-\varphi$? But once we substitute $\psi$ by $u-\varphi$, we will introduce $\varphi$......Does this the author's error?