Exam
UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph. D. Preliminary Examination in Differential Equations August 15, 2011.
Fall 2011 · 11 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Let \(f(t,x)\) be continuously differentiable for \((t,x)\in \mathbb {R}\times \mathbb {R}^n\) and let \(x_0\in \mathbb {R}^n\) be a point. Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([-\epsilon ,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]
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Question 2
Suppose \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\) and periodic of period \(T>0\). Consider the linear differential equation in \(\mathbb {R}^n\) \begin{equation} \begin {cases} \dfrac {dx}{dt}=A(t)x,\\ x(0)=x_0. \end {cases} \tag {1} \end{equation} Let \(\Phi (t)\) be the fundamental matrix solution with \(\Phi (0)=I\).
- Define: Floquet Matrix, Floquet Multiplier and Floquet Exponent. How are these related to \(\Phi (t)\)? State the necessary and sufficient conditions on (1) so that all solutions are bounded.
- Prove that all solutions of the Mathieu Equation (2) are bounded \begin{equation} \ddot {y}+\left (1+\frac {1}{20}\sin 4t\right )y=0. \tag {2} \end{equation}
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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 \(f(t,x)\) be a \(C^2\) function such that \(\lvert f(t,x)\rvert =o(\lvert x\rvert )\) as \(x\to 0\), uniformly in \(t\). Consider the initial value problem for \(x_0\in \mathbb {R}^n\), \begin{equation} \begin {cases} \dfrac {dx}{dt}=Ax+f(t,x),\\ x(0)=x_0. \end {cases} \tag {3} \end{equation}
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Show that if \(\lvert x_0\rvert \) is small enough, there is a bounded solution to (3) that exists for all time.
[Hint: Suppose \(g(t)\) and \(u(t)\) are nonnegative functions and \(c_0\ge 0\) is a constant that satisfy \(u(t)\le c_0+\int _0^t g(s)u(s)\dd s\) for all \(t\ge 0\). Then Gronwall’s Inequality implies \[ u(t)\le c_0\exp \left (\int _0^t g(s)\dd s\right ) \] for all \(t\ge 0\).]
- Let \(z(t)=0\) for all \(t\) be the zero solution. Define what it means for \(z\) to be Lyapunov Stable (some authors call this notion stable .) Show that \(z\) is Lyapunov stable for this equation.
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Question 4
A nonconstant periodic orbit near the origin is created or destroyed as the parameter \(\mu \) passes through zero in the one parameter family of equations \[ \begin {aligned} \dot {x} &= y+\mu x,\\ \dot {y} &= -x+\mu y-x^2y. \end {aligned} \]
- Determine whether the origin is stable or unstable for the system when \(\mu =0\).
- For which values of \(\mu \) near zero does the system have a nonconstant periodic orbit near the origin? Is this orbit attracting or repelling? Why?
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Question 5
Consider a modified predator-prey system for populations \(x,y\) given by \[ \begin {aligned} \dot {x} &= x\cdot \left (1-\frac {x}{4}-\frac {y}{1+x}\right ),\\ \dot {y} &= y\cdot \left (\frac {x}{1+x}-\frac 12\right ) \end {aligned} \]
- Show that the triangle with vertices \((0,0)\), \((5,0)\) and \((0,5)\) is a forward invariant subset.
- Find the fixed points and determine their stability.
- Show that the system has at least one nonconstant periodic solution.
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Question 6
Suppose that \(f,g:\mathbb {R}^3\to \mathbb {R}\) are smooth functions. Consider the parameter dependent system \begin{equation} \begin {aligned} \dot {x} &= f(x,y;\epsilon ),\\ \dot {y} &= g(x,y;\epsilon ). \end {aligned} \tag {4} \end{equation}
- Suppose \(\Gamma _0\) is a nonconstant \(T\)-periodic solution of (4) when \(\epsilon =0\). Define Poincarè Section, Poincaré Map for \(\Gamma _0\).
- Find a nontrivial periodic solution \(\Gamma _0\) for the system (5) when \(\epsilon =0\). Show that a nontrivial periodic solution \(\Gamma _\epsilon \) for the \(\epsilon \)-system can be continued from \(\Gamma _0\) as the parameter varies through small \(\lvert \epsilon \rvert \). \begin{equation} \begin {aligned} \dot {x} &= y+x(1-x^2-y^2),\\ \dot {y} &= -x+y(1-x^2-y^2)+\epsilon y. \end {aligned} \tag {5} \end{equation}
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Question 7
Assume that \(u\in C^2(\mathbb {R})\), and \(f\in C_0^2(\mathbb {R})\).
- Assume \(u\) satisfies \(u''=0\). Without using an explicit representation for the solution, show that the mean value property \[ u(x)=\frac 12\bigl (u(x+r)+u(x-r)\bigr )=\frac {1}{2r}\int _{x-r}^{x+r}u(t)\dd t, \] for each interval \((x-r,x+r)\), is satisfied, and conversely show that the mean value property implies \(u''=0\).
- Assuming \(u''=0\) in the bounded interval \((a,b)\), use part (a) to prove the strong maximum principle for \(u\).
- Find a radially symmetric fundamental solution \(\Phi (x)\) for the problem \(u''=0\), and use it to prove that \[ u(x)=\int _{-\infty }^{\infty }\Phi (x-y)f(y)\dd y \] satisfies \(-u''=f\) in \(\mathbb {R}\).
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Question 8
Given \(f\in C^\infty (\mathbb {R}^n)\), and \(g\in C^\infty (\mathbb {R}^{n-1})\), consider the eikonal equation \[ \lvert Du\rvert =f(x),\qquad \text {in }\mathbb {R}^n, \] with the boundary condition \(u=g\) on the plane \(\Gamma =\{x_n=0\}\).
- Write down the characteristic equations \[ \begin {aligned} \dot {p} &= -D_xF-D_zF\,p,\\ \dot {z} &= D_pF\cdot p,\\ \dot {x} &= D_pF, \end {aligned} \] associated with this problem.
- Given an initial point \(x^0\in \Gamma \), if possible, determine a set of initial conditions for the characteristic equations which satisfy the compatibility conditions (in other words, the initial conditions are admissible).
- Determine conditions on \(g\) and \(f\) which guarantee a unique local solution to the boundary value problem near \(x^0\).
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Question 9
Compute the unique entropy solution of \[ \begin {aligned} u_t+\left (\frac {u^2}{2}\right )_x &= 0 && \text {in }\mathbb {R}\times (0,\infty ),\\ u &= g && \text {on }\mathbb {R}\times \{0\}, \end {aligned} \] where \[ g(x)= \begin {cases} 0, & \text {if }x<0,\\ x, & \text {if }0\le x\le 1,\\ 0, & \text {if }x>1. \end {cases} \] Draw a sketch documenting your answer, illustrating what happens at all times \(t>0\).
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Question 10
Let \(u\) solve the initial value problem \[ \begin {aligned} u_{tt}-u_{xx} &= 0, && \text {in }\mathbb {R}\times (0,\infty ),\\ u=g,\quad u_t=h, &&& \text {on }\mathbb {R}\times \{0\}. \end {aligned} \] Suppose \(g,h\) have compact support. Define the kinetic energy \(k(t)=\frac 12\int _{-\infty }^{\infty }u_t^2(x,t)\dd x\) and the potential energy \(p(t)=\frac 12\int _{-\infty }^{\infty }u_x^2(x,t)\dd x\). Prove
- \(k(t)+p(t)\) is constant in \(t\).
- \(k(t)=p(t)\) for all \(t\) sufficiently large.
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Question 11
Let \(a(x)\in C^\infty (\overline {U})\), where \(U\) is a smooth bounded connected domain, \(U\subset \mathbb {R}^n\). Assume \(a(x)\) is uniformly bounded away from zero on \(U\). Consider the initial-boundary value problem \[ (*)\quad \begin {cases} u_t=\operatorname {div}(aDu), & \text {in }U\times (0,\infty ),\\ u=0, & \text {on }\partial U\times (0,\infty ),\\ u=g, & \text {on }U\times \{0\}. \end {cases} \]
- If \(g\) minimizes \(\lambda =\int _U a\lvert Dg\rvert ^2\dd x\) over all \(g\in H_0^1(U)\) satisfying \(\int _U g^2\dd x=1\), characterize \(\int _U u(x,1)^2\dd x\) in terms of \(\lambda \).
- Let \(J(g)=\int _U u(x,1)^2\dd x\), where \(u\) solves \((*)\) with initial condition \(g\), and denote \[ B=\{g\in L^2(U):\norm {g}_2=1\}. \] Denote by \(g_1\) the minimizer of \(\lambda \) from part (a). Prove \[ \sup _{g\in B} J(g) \] is attained by \(g_1\). Does \(J(g)\) have a minimizer over \(B\)? What is \(\inf _{g\in B}J(g)\)?