UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph. D. Preliminary Examination in Differential Equations August 16, 2010

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} \frac {dx}{dt}=A(t)x \tag {1} \end{equation} Let \(\Phi (t)\) be the fundamental matrix solution with \(\Phi (0)=I\).

1. Let \(\zeta (t)\in C^1([0,\infty ),\mathbb {R}^n)\) be a solution of (1). Define: \(\zeta (t)\) is a Stable solution. (This notion is also sometimes called Liapunov Stable.)
2. Show that the zero solution of (1) is stable if and only if every eigenvalue \(\omega \) of \(\Phi (T)\) satisfies either \(\lvert \omega \rvert <1\), or \(\lvert \omega \rvert =1\) but in this case \(\omega \) appears only in matrices \(J_i\) (from the Jordan Canonical Form for \(\Phi (T)\)) such that \(J_i\) is a \(1\times 1\) matrix.

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 \(B(t)\) be a smooth real \(n\times n\) matrix function and \(\beta >0\) such that \(\lvert B(t)\rvert \le e^{-\beta t}\) for all \(t\ge 0\). Consider the initial value problem for \(x_0\in \mathbb {R}^n\), \begin{equation} \begin {aligned} \frac {dx}{dt} &= [A+B(t)]x,\\ x(0) &= x_0. \end {aligned} \tag {2} \end{equation}

1. Show that the solution to (2) exists for all time and \(\lim _{t\to \infty }x(t)=0\).

   [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 \(t\ge 0\).]

2. Let \(z(t)=0\) for all \(t\) be the zero solution. Define what it means for \(z\) to be Asymptotically Stable. Show that \(z\) is asymptotically stable for this equation.

Suppose that \(f:\mathbb {R}^2\times \mathbb {R}\to \mathbb {R}^2\) is a smooth function and \(\Gamma _0\subset \mathbb {R}^2\) is a nonconstant periodic orbit of the parameter dependent autonomous system for the parameter value \(\epsilon =\epsilon _0\), \begin{equation} \frac {dx}{dt}=f(x,\epsilon _0). \tag {3} \end{equation}

1. Define the following: \(\Sigma \) is a Poincaré section for for \(\Gamma _0\) at the point \(x_0\in \Gamma _0\) and \(P:\Sigma \to \Sigma \) is the Poincaré first return map.

2. Define what it means for the orbit \(\Gamma _0\) to be hyperbolic. Your definition should make the following theorem true.

   Theorem 1. Suppose that the \(\Gamma _0\) is a hypererbolic nonconstant periodic orbit of the system (3). Then there is an \(\eta >0\) depending on \(f\) and \(\Gamma _0\) and a continuous one-parameter family of closed curves \(\Gamma _\epsilon \) such that \(\Gamma _0=\Gamma _{\epsilon _0}\) and that \(\Gamma _\epsilon \) are nonconstant periodic orbits of the perturbed systems \[ \frac {dx}{dt}=f(x,\epsilon ) \] for all \(\epsilon \) such that \(\lvert \epsilon -\epsilon _0\rvert <\eta \).

3. Sketch a proof of Theorem 1.
4. Show that the \(\epsilon =1\) solution \((x=\cos t)\) can be continued to a one parameter family of solutions in a neighberhood of \(\epsilon =1\) by checking that the conditions of the theorem are satisfied: \[ \ddot {x}+(x^2+\dot {x}^2-\epsilon )\dot {x}+x=0. \]

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}

1. Define what it means to be a weak solution of (4).
2. 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.

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 Dirichlet boundary conditions (\(\phi =0\) on \(\partial D\)) and with eigenvalue \(\lambda \). \begin{equation} \Delta \phi +\lambda \phi =0. \tag {5} \end{equation}

1. 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 \[ \int _D u^2 \le C\int _D \lvert Du\rvert ^2. \]
2. By integrating eigenfunctions of (5), conclude that eigenvalues are positive \(\lambda >0\).
3. 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 \(\lambda _1(D_1)\ge \lambda _1(D_2)\).

Let \(D\in \mathbb {R}^d\) be a bounded domain with smooth boundary \(\partial D\). Assume that that a complete orthonormnal set of eigenfunctions \(\{\phi _n\}_{n=1}^\infty \) for the Laplacian on \(D\) with Dirichlet boundary conditions are known, in other words \begin{equation} -\Delta \phi _n=\lambda _n\phi _n,\qquad \text {in }D,\ n=1,2,3\ldots , \tag {6} \end{equation} with \(\phi _n(x)=0\) for \(x\in \partial D\), and \(\int _D \phi _n^2\dd x=1\).

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 \[ \begin {aligned} -u_t+\Delta u &= 0, && t>0,\ x\in D,\\ u(x,t) &= 0, && x\in \partial D,\\ u(x,0) &= 0, && x\in D. \end {aligned} \]
2. Show that there is a constant \(c\) such that \[ \left \lvert \int _D u(x,t)\phi _n(x)\dd x\right \rvert \le c e^{-\lambda _n t} \] for all \(n\). Find \(\lim _{t\to \infty }u(x,t)\).
3. Give an example (the case \(d=1\) will suffice) which shows that solutions of the heat equation do not exhibit finite propagation speed.