PDE - Energy - Wave Equation I dont know how solve a) and b), I'm read the book of Walter Strauss, but I have a lot of doubts, for the c) first, I tried estimate $(k(t)e^{-2at})'$ and integrate the inequal, but not unwind... :( Let $a\in \mathbb{R}$ and consider the solution of the equation $$u_{tt}+au_{t}-u_{xx} =0\;,\;t>0\;,\;x\in(0,1) \\\ u(0,t)=u(1,t)=0,\; t>0 \\\ u(x,0)= \varphi(x)\;,u_t(x,0)=\psi(x)\; ,\;x \in (0,1) $$ Define the energy associated by $$E(t):= \frac{1}{2}\int_0^1(u_t^2(x,t)+u_x^2(x,t))dx. $$ a) show that $$E(t)=-ak(t)+E(0)$$ where $k(t)=\int_0^t\int_0^1u_t^2(x,s)dxds$ b) show that $$k'(t)\leq -2ak(t)+2E(0)$$ c) shows the estimated decay for $k$: $$k(t)\leq\frac{E(0)}{a}(1-e^{-2at})$$

This solves (a) and should get you started for the other questions. Note that $u_{tt}=u_{xx}-au_t$ hence $$E'(t)=\int_0^1 (u_tu_{tt}+u_xu_{xt})\mathrm dx=\int_0^1 (u_tu_{xx}+u_xu_{xt})\mathrm dx-a\int_0^1 (u_t)^2\mathrm dx.$$ The first term on the RHS is $$\int_0^1 (u_tu_x)_x\mathrm dx=u_t(1,t)u_x(1,t)-u_t(0,t)u_x(0,t)=0,$$ since $u(0,t)$ and $u(1,t)$ do not depend on $t$ hence $u_t(0,t)=u_t(1,t)=0$. Integrating $E'(t)$ from $0$ to $t$ yields the desired formula since the remaining term on the RHS is $-ak'(t)$.