Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2011 • Question 7

Question

Math 6410 • ODE Spring 2011 • Question 7

Suppose that \(u\in C^2\) solves \[ u_{tt}-\operatorname {div}(aDu)=0,\qquad \text {in }\mathbb {R}^n\times (0,\infty ), \] where \(a\in C^1(\mathbb {R}^n)\) satisfies \(\frac 12\le a(x)\le 1\) for all \(x\).

Assunimg \(\operatorname {supp}_x u(x,t)\subset B(0,R)\subset \mathbb {R}^n\) for all \(0\le t\le T\), show that the energy \[ E(t)=\frac 12\int _{\mathbb {R}^n}u_t^2+a\lvert Du\rvert ^2\dd x,\qquad 0\le t\le T, \] is conserved.
Using an appropriately defined local energy, establish that solutions exhibit finite propagation speed: if for some \(r>0\), \(u(x,0)\) and \(u_t(x,0)\) are zero for \(x\in B(x_0,r)\subset \mathbb {R}^n\), then \(u\) is zero in the cone \(K=\{(x,t):0\le t\le r,\lvert x-x_0\rvert <r-t\}\).

Suppose that \(u\in C^2\) solves
\[
  u_{tt}-\operatorname{div}(aDu)=0,\qquad \text{in }\R^n\times(0,\infty),
\]
where \(a\in C^1(\R^n)\) satisfies \(\frac12\le a(x)\le1\) for all \(x\).
\begin{enumerate}
\item[(a)] Assunimg \(\operatorname{supp}_x u(x,t)\subset B(0,R)\subset\R^n\) for all
\(0\le t\le T\), show that the energy
\[
  E(t)=\frac12\int_{\R^n}u_t^2+a\lvert Du\rvert^2\dd x,\qquad 0\le t\le T,
\]
is conserved.
\item[(b)] Using an appropriately defined local energy, establish that solutions
exhibit finite propagation speed: if for some \(r>0\), \(u(x,0)\) and \(u_t(x,0)\) are zero
for \(x\in B(x_0,r)\subset\R^n\), then \(u\) is zero in the cone
\(K=\{(x,t):0\le t\le r,\lvert x-x_0\rvert<r-t\}\).
\end{enumerate}

Topic Tags

Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Finite Speed of Propagation and Domain of Dependence Poincare Maps and Periodic Solutions