Exam
UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS
Fall 2025 · 6 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Let \(U\) be a bounded domain in \(\mathbb {R}^d\) and \(\Gamma \) be a closed subset of \(\partial U\). Consider the energy
\[ I[v]=\frac 12\int _U \left (|Dv|^2+u^2\right )\,dx \]on the admissible class
\[ \mathcal {A}=\{v\in C^2(\overline {U})\cap C^1(\overline {U}): u|_{\Gamma }=g\},\quad g\in C(\partial U). \]Suppose that \(u\) is a minimizer for the calculus of variations problem
\[ I[u]=\min _{v\in \mathcal {A}} I[v] \]find and justify the PDE boundary value problem solved by \(u\). Show that there is at most one solution of this PDE in the admissible class \(\mathcal {A}\).
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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 \((t_0,x_0)\).
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Question 3
Let \(U\) be a bounded domain and \(u\) solve the heat equation \[ \begin {cases} \partial _t u=\Delta u+cu & \text {in } U\times (0,\infty ),\\ u(x,0)=g(x),\\ u(x,t)=0 & \text {on } \partial U\times (0,\infty ). \end {cases} \]
Classify, with justification, the optimal range of values for \(c\in \mathbb {R}\) so that for any smooth initial data \(g\) the solution \(u\) above converges to 0 as \(t\to \infty \).
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Question 4
Find the entropy solution of the following scalar conservation law, arising in traffic flow modeling,
\[ \begin {cases} \partial _t u+\partial _x(u(1-u))=0 & \text {for } x\in \mathbb {R},\ t>0,\\ u(x,0)=g(x) \end {cases} \]with initial data
\[ g(x)= \begin {cases} 0, & x<-1,\\ 1, & -1<x<0,\\ 0, & 0<x<1,\\ \frac 12, & 1<x. \end {cases} \]After the rarefaction hits the shock you should write the explicit ODE IVP for the shock location \(\gamma (t)\) and a piecewise definition of the solution using this shock location \(\gamma (t)\), but you don’t actually need to solve the ODE IVP to get the explicit formula for \(\gamma (t)\).
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Question 5
Let \(f:\mathbb {R}^d\to \mathbb {R}\) smooth. Show that there is at most one smooth solution of \[ \begin {cases} -\Delta u=f & \text {in } \mathbb {R}^d,\\ \displaystyle \lim _{|x|\to \infty }u(x)=0. \end {cases} \]
Give a (non-trivializing) additional condition on \(f\) so that you can show existence of such a solution.
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Question 6
Suppose that \(U\subset \mathbb {R}^n\) is a bounded domain and \(b:U\to \mathbb {R}^n\) is a smooth velocity field. Suppose that \(u(x, t)\) solves the reaction-diffusion- advection equation with a logistic reaction term
\[ \begin {cases} \partial _t u+b(x)\cdot \nabla u-\Delta u=u(1-u) & \text {in } U\times (0,\infty ),\\ u=0 & \text {on } \partial U\times [0,\infty ). \end {cases} \]Show that if \(u(x, 0) < 1\) in \(U\) then \(u(x, t) < 1\) for all \(t > 0\) and all \(x \in U\).