Problems to Section 2.7


### Problems

Problem 1. For equation $$u_{tt}-c^2 u_{xx}+ f(u)=0, \qquad x> 0$$ prove energy conservation law $$E(t)=\frac{1}{2}\int_0^\infty \bigl(u_t^2+ c^2 u_{x}^2+F(u)\bigr)\,dx$$ under Dirichlet ($u|_{x=0}=0$) or Neumann ($u_x|_{x=0}=0$) boundary condition; here $F$ is a primitive of $f$.

Problem 2. For beam equation $$u_{tt}+K u_{xxxx}=0, \qquad x> 0,\\$$ prove energy conservation law $$E(t)=\frac{1}{2}\int_0^\infty (u_t^2+ K u_{xx}^2)\,dx$$ under each of the pair of the boundary conditions: \begin{align} &u|_{x=0}=u_x |_{x=0}=0;\\ &u|_{x=0}=u_{xxx} |_{x=0}=0;\\ &u_{x}|_{x=0}=u_{xx} |_{x=0}=0. \end{align}

Problem 3. a. For problem \begin{align} &u_{tt}-c^2 u_{xx}=0, &&x >0,\\ &(\alpha_0 u_x+\alpha_1 u)|_{x=0}=0 \end{align} find energy conservation law $$E(t)=\frac{1}{2}\int_0^\infty (u_t^2+ c^2 u_{x}^2)\,dx + \frac{1}{2}a u(0)^2$$ (you need to calculate $a$).

1. For problem \begin{align} &u_{tt}-c^2 u_{xx}=0, &&0< x < l,\\ &(\alpha_0 u_x+\alpha_1 u)|_{x=0}=0,\\ &(\beta_0 u_x-\beta_1 u)|_{x=l}=0 \end{align} find energy conservation law $$E(t)=\frac{1}{2}\int_0^\infty (u_t^2+ c^2 u_{x}^2)\,dx + \frac{1}{2}a u(0)^2+ \frac{1}{2}b u(l)^2$$ (you need to calculate $a$, $b$).

Problem 4. For problem \begin{align} &u_{tt}-c^2 u_{xx}=0, &&x >0,\\ &(u_x-i\alpha u_t)|_{x=0}=0 \end{align} with real $\alpha$ prove energy conservation law $$E(t)=\frac{1}{2}\int_0^\infty (|u_t|^2+ c^2 |u_{x}|^2)\,dx$$

Problem 5. Consider Problem 2.6.2. \begin{align*} & u_{tt}-c_1^2u_{xx}=0, &&&t>0, x>0, \\ & u_{tt}-c_2^2u_{xx}=0, &&&t>0, x<0, \\ &u|_{t=0}= \phi (x), &&u_t|_{t=0}= c_1\phi'(x) &x>0, \\ &u|_{t=0}=0, &&u_t|_{t=0}=0, &x<0,\\ &u|_{x=+0}=\alpha u|_{x=-0}, &&u_x|_{x=+0}=\beta u_x|_{x=-0} &t>0 \end{align*}

Let $E_1(t)=\frac{m_1}{2}\int_0^\infty (u_t^2 + c_1^2 u_x^2)\,dx$, $E_2(t)=\frac{m_2}{2}\int_{-\infty}^0 (u_t^2 + c_2^2 u_x^2)\,dx$.

1. Find $m_1:m_2$ such that $E(t)=E_1(t)+E_2(t)$ is conserved.
2. In this case prove that if $\phi (x)=0$ for $x> L$ then for $t > L/c_1$ $E_1(t)= k_1 E(0)$, $E_2(t)= k_2 E(0)$. Calculate $k_1,k_2$ and prove that $k_1+k_2=1$.