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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2025 • Question 2

Question

Math 6420 • PDE Fall 2025 • Question 2

Let \(c>0\) and \(\mu (x)\) a smooth function. Consider the wave equation

\begin{equation} \tag {1} \begin {cases} \partial _t^2 u-c^2\Delta u+\mu (x)\partial _t u=f & \text {in } (0,T]\times \mathbb {R}^d,\\ (u,u_t)=(\varphi ,\psi ) & \text {on } \{t=0\}\times \mathbb {R}^d. \end {cases} \end{equation}

Explain the notion of domain of dependence in the context of wave-type PDEs. Show finite speed of propagation for this PDE by showing, using an energy method, a natural upper bound on the domain of dependence of some point \((t_0,x_0)\).

Raw LaTeX

Let \(c>0\) and \(\mu(x)\) a smooth function. Consider the wave
equation

\begin{equation}
\tag{1}
\begin{cases}
\partial_t^2 u-c^2\Delta u+\mu(x)\partial_t u=f & \text{in } (0,T]\times\mathbb{R}^d,\\
(u,u_t)=(\varphi,\psi) & \text{on } \{t=0\}\times\mathbb{R}^d.
\end{cases}
\end{equation}

Explain the notion of domain of dependence in the context of wave-type
PDEs. Show finite speed of propagation for this PDE by showing, using
an energy method, a natural upper bound on the domain of dependence of
some point \((t_0,x_0)\).

Topic Tags

Distributions and Weak Solutions Finite Speed of Propagation and Domain of Dependence Fundamental Solutions Wave Equation and Energy Methods