Let $y$ be the solution of the following nonlinear wave equation $$ \begin{cases} \begin{array}{ll} y_{tt}-\Delta y + f(y) = F & (t,x)\in (0,T)\times(0,1)\\ y(.,0)=y(.,1)=0 & t\in(0,T)\\ y(0,.)=y^0 \in H^1_0(0,1), \,\, y_t(0,.)=y^1\in L^2(0,1) & x\in (0,1). \end{array} \end{cases} $$ Where $f \in C^1(\mathbb{R})$ the nonlinearity, and $F \in L^{2}((0,T)\times(0,1))$.
I want to obtain an estimate of the following term $||y(T,.)||_{H^1_0(0,1)}$ and $||y_t(T,.)||_{L^2(0,1)}$, in terms of $||y(0,.)||_{H^1_0(0,1)}$ and $||y_t(0,.)||_{L^2(0,1)}$ and probably $||y_{tt}-\Delta y + Ay||_{L^2((0,T)\times(0,1))}$.
My idea is that we can obtain such estimation (but only for the linear case): $$ \begin{cases} \begin{array}{ll} y_{tt}-\Delta y + Ay = F & (t,x)\in (0,T)\times(0,1)\\ y(.,0)=y(.,1)=0 & t\in(0,T)\\ y(0,.)=y^0 \in H^1_0(0,1), \,\, y_t(0,.)=y^1\in L^2(0,1) & x\in (0,1). \end{array} \end{cases} $$ Where $A\in L^{\infty}((0,T)\times (0,1))$ and $F \in L^{2}((0,T)\times(0,1))$.
is the estimation of the enegry defined as follows: $$E(t)= ||y(t)||^2_{H^1_0(0,1)}+||y_t(t)||^2_{L^2(0,1)}$$ and we have the following $$E(T)\leq C.E(0)e^{C\sqrt{||A||_{\infty}}}$$ for some $C>0$.
I'm stuck on the nonlinearity.