Exam
UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph. D. Preliminary Examination in Differential Equations January 7, 2009
Spring 2009 · 12 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Suppose \(f(x,t)\in C^1(\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) and \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Show that for the initial value problem \begin{equation} \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(t_0)=x_0. \end {cases} \tag {1} \end{equation} there exists a unique local solution, i.e., for some \(\epsilon >0\) there is a unique function \(y\in C^1((t_0-\epsilon ,t_0+\epsilon ),\mathbb {R}^n)\) which satisfies (1) for \(t\in (x_0-\epsilon ,x_0+\epsilon )\).
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Question 2
Consider the \(T\)-periodic non-autonomous linear differential equation in \(\mathbb {R}^n\) \begin{equation} \frac {dx}{dt}=A(t)x \tag {2} \end{equation} where \(A(t)\) is a real, smooth \(n\times n\) matrix function such that \(A(t+T)=A(T)\) for all \(t\). Let \(\Phi (t)\) be the fundamental matrix with \(\Phi (0)=I\).
- Show that there is a \(T\)-periodic function \(P(t)\) and a matrix \(B\) so that \[ \Phi (t)=P(t)e^{tB}. \]
- Suppose the eigenvalues \(\omega \) of \(\Phi (T)\) satisfy \(\lvert \omega \rvert \le 1\) and that all \(\omega \) with \(\lvert \omega \rvert =1\) possess a full set of eigenvactors. Show that then all solutions of (2) are bounded.
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Question 3
Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda _i\) satisfy \(\Re \lambda _i<0\) for all \(i=1,\ldots ,n\). Let \(g(x,t)\in C^\infty (\mathbb {R}^n\times \mathbb {R},\mathbb {R}^n)\) such that \(\lvert g(x,t)\rvert \le \lvert x\rvert ^2\) for all \(x\in \mathbb {R}^n\). Consider the ODE \[ y'=Ay+g(y,t). \]
- Show that if the initial value \(\lVert x(0)\rVert \) is small enough, then the solution exists for all time and \(\lim _{t\to \infty }x(t)=0\). [Hint: Gronwall’s Inequality for nonnegative functions \(u(t)\) and \(v(t)\) that satisfy \(v(t)\le c_0+\int _0^t u(s)v(s)\dd s\) is \(v(t)\le c_0\exp \left (\int _0^t u(s)\dd s\right )\).]
- Let \(z(t)=0\) for all \(t\) be the zero solution. Define what it means for \(z\) to be Stable (often called Liapunov Stable.) Discuss the stability of \(z\) for this equation.
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Question 4
Consider the scalar equation \[ \ddot {u}+\epsilon (u^2-1)\dot {u}+u=0,\qquad 0<\epsilon \ll 1. \] Apply the method of multiple scales, with variables \(t\) and \(\eta =\epsilon t\) to show that if \(u(0)=a>0\) and \(\dot {u}(0)=0\) then for \(t=O(\epsilon ^{-1})\), \[ u=\frac {2\cos t}{\sqrt {1+\left (\frac {4}{a^2}-1\right )e^{-\epsilon t}}} +O(\epsilon ). \]
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Question 5
Consider the scalar differential equation \begin{equation} \ddot {u}-\epsilon u^3+u=0,\qquad 0<\epsilon \ll 1. \tag {3} \end{equation}
- Suppose that (3) has a \(T\)-periodic solution \(u(t;\epsilon )\). Let \(\mathcal {E}=\frac 12(u^2+\dot {u}^2)\). Using the fact that \(\mathcal {E}(T)=\mathcal {E}(0)\), find the amplitude \(u(0;\epsilon )=a(\epsilon )\) with \(\dot {u}(0;\epsilon )=0\) and period \(T(\epsilon )\) for an approximate periodic solution.
- Determine the stability of the periodic solution.
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Question 6
Show that the system \[ \begin {aligned} \dot {x}&=x+y-x^3\\ \dot {y}&=-x+y-y^3 \end {aligned} \] has at least one nonzero periodic solution.
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Question 7
Consider the scalar first order PDE \begin{equation} \begin {aligned} uu_x+u_y&=0,\\ u(x,0)&=h(x). \end {aligned} \tag {4} \end{equation}
- Define what it means to be a weak solution of (4).
- Suppose that \(u_0>0\) and \[ h(x)= \begin {cases} u_0, & \text {if }x\le 0,\\ u_0(1-x), & \text {if }0<x<1,\\ 0, & \text {if }x\ge 1. \end {cases} \] Show that a shock develops at a finite time and describe the global weak solution.
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Question 8
Traffic flow may be modeled by the equation \[ \frac {\partial }{\partial x}\bigl (G(\rho )\bigr )+\frac {\partial }{\partial t}\rho =0 \] where \(\rho (x,t)\) is the density of traffic flow along a highway. If \(\rho _{\max }\) is the maximum density of cars in a bumper to bumper situation, then \[ G(\rho )=c\rho \left (1-\frac {\rho }{\rho _{\max }}\right ) \] where \(c>0\) is the free speed constant. Suppose that the initial concentration is given by \[ \rho (x,o)= \begin {cases} \frac 12\rho _{\max }, & \text {if }x<0,\\ 0, & \text {if }x\ge 0. \end {cases} \] Find the shock curve and describe the weak solution. Interpret your result in terms of traffic flow.
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Question 9
Let \(D\in \mathbb {R}^n\) be a bounded, connected domain with smooth boundary. Let \(\phi \) be an eigen-funcion of the Laplacian on \(D\) with with Dirichlet boundary conditions (\(\phi =0\) on \(\partial D\)) and with eigenvalue \(\lambda \), \begin{equation} \Delta \phi +\lambda \phi =0. \tag {5} \end{equation}
- Show that there is a finite constant \(C\) so that for every nonzero continuously differentiable \(u\in C^1(D)\) such that \(u=0\) on \(\partial D\) satisfies \[ \int _D u^2\le C\int _D \lvert Du\rvert ^2. \]
- By integrating eigenfunctions of (5), conclude that eigenvalues are positive \(\lambda >0\).
- Denote by \[ \lambda _1(D)= \inf _{\substack {u\in C_0^1(D),\ u\not \equiv 0}} \frac {\int _D \lvert Du\rvert ^2}{\int _D u^2} \] the least value of the Rayleigh quotient (which is at least \(1/C\)). Assuming that \(\lambda _1(D)\) is an eigenvalue of \(D\) with Dirichlet boundary conditions, show that if two such domains satisfy \(D_1\subset D_2\), then \(\lambda _1(D_1)\ge \lambda _1(D_2)\).
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Question 10
Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\quad x\in \mathbb {R}. \] along with the intial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]
- Assuming \(c\) is constant, derive d’Alembert’s formula \[ u(x,t)=\frac 12\bigl (g(x+ct)+g(x-ct)\bigr ) +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
- Briefly explain the concepts of domain of dependence, and finite propagation speed, and prove that these properties follow from d’Alembert’s formula.
- Suppose that we restrict the spatial domain to \(x>0\) and impose the boundary condition \(u(0,t)\) for \(t\ge 0\). If the initial conditions satisfy \(g(0)=h(0)=0\), using d’Alembert’s formula as a starting point, find a closed-form solution for \(u(x,t)\), with \(x,t>0\).
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Question 11
Let \(D\subset \mathbb {R}^d\) be a bounded domain with smooth boundary \(\partial D\). Assume that the complete, orthonormal set of eigenfunctions \(\{\phi _n\}_{n=1}^{\infty }\) for the negative Dirichlet Laplacian on \(D\) is known, in other words, \[ -\Delta \phi _n=\lambda _n\phi _n,\quad \text {in }D,\ n=1,2,3,\ldots , \] with \(\phi _n(x)=0\) for \(x\in \partial D\), and \(\int _D\phi _n^2\dd x=1\).
- Given \(f\in L^2(D)\), find a closed-form series solution \(u(x,t)\) for the initial boundary value problem for the heat equation \[ \begin {aligned} u_t-\Delta u&=0,\quad t>0,\ x\in D\\ u(x,t)&=0,\quad x\in \partial D,\\ u(x,0)&=f,\quad x\in D. \end {aligned} \]
- Show that there exists a constant \(C\) such that \(\left \lvert \int _D u(x,t)\phi _n(x)\dd x\right \rvert \le Ce^{-\lambda _n t}\), for all \(n\). Find \(\lim _{t\to 0}u(x,t)\).
- Give an example (the case \(d=1\) will suffice) which shows that the heat equation does not exhibit finite propagation speed.
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Question 12
Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).
- Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\) (the ball of radius \(r\), centered at \(y\)) satisfies \(B_r(y)\subset \Omega \), then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi } u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
- Assuming \(\Omega \) is connected, prove that \(u\) can attain a maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.