UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations August 19, 2013.

Let \(f(x,t)\) be continuous for all \((x,t)\in \mathbb {R}^n\times \mathbb {R}\) and let \((x_0,t_0)\in \mathbb {R}^n\times \mathbb {R}\). Suppose that there is \(M\in \mathbb {R}\) such that \[ \norm {f(x,t)-f(y,t)}\le M\norm {x-y}, \qquad \text {for all }(x,t),(y,t)\in \mathbb {R}^n\times \mathbb {R}. \] Show that there is and \(\epsilon >0\) and a unique function \(y\in C^1([0,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x,t),\\ x(0)=x_0. \end {cases} \]

Let \(A\) be an \(n\times n\) real matrix. Consider the linear initial value problem \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] Give an algebraic condition on the matrix \(A\) which holds if and only if all solutions of the initial value problem are bounded for all \(t\in \mathbb {R}\). Prove the equivalence.

Consider a nonconstant \(T\)-periodic orbit \(\gamma (t)\) of the smooth autonomous differential equation \[ \dot {x}=f(x),\qquad x\in \mathbb {R}^n. \]

1. Describe the Poincaré Return Map associated to \(\gamma \).
2. A \(2\pi \)-periodic solution of the planar system \[ \begin {aligned} \dot {x} &= x-(x+y)\sqrt {x^2+y^2},\\ \dot {y} &= y+(x-y)\sqrt {x^2+y^2} \end {aligned} \] is given by \(\gamma (t)=(\cos t,\sin t)\). Find a Poincaré Return Map associated to \(\gamma \).
3. Find the derivative of the Poincaré Return Map from part (b) at \(\gamma \).
4. Using (c), determine whether the solution (b) is stable and define the sense of stability you mean.

Consider a modified predator-prey model in the first quadrant. Show that it has at least one nonconstant periodic solution. \[ \begin {aligned} \dot {x} &= x(8-x)-\frac {9xy}{x+1},\\ \dot {y} &= y\left (1-\frac {y}{x}\right ) \end {aligned} \]

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Using an approximate center manifold, determine the nature of the bifurcation from \((x,\dot {x},\epsilon )=(0,0,0)\) for the quadratic Duffing’s Equation \[ \ddot {x}+\dot {x}+\epsilon x+x^2=0. \]

The small longitudinal free vibrations of an elastic bar are governed by the following equation: \[ \rho (x)\frac {\partial ^2u}{\partial t^2} =\frac {\partial }{\partial x}\left [E(x)\frac {\partial u}{\partial x}\right ]. \] Here \(u\) is the longitudinal displacement, \(\rho \) is the linear density of the material and \(E\) is its Young’s modulus. Assume that the bar is constructed by welding together two bars of different (constant) Young’s moduli \(E_1,E_2\) and densities \(\rho _1,\rho _2\), respectively. The displacement \(u\) is continuous across the junction, which is located at \(x=0\).

1. Give a weak formulation of the global initial value problem, and use this to derive the following jump condition: \[ E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0. \]
2. Let \(c_j^2=E_j/\rho _j\). A left incoming wave \(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave \(u_R(x,t)=R e^{i(t+x/c_1)}\) and a transmitted wave \(u_T=T e^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

Consider the following inhomogeneous initial-Neumann problem: \[ \begin {aligned} u_t &= Du_{xx}+\alpha tx, && 0<x<\pi ,\quad 0<t,\\ u(x,0) &= 1, && 0<x<\pi ,\\ u_x(0,t)&=u_x(\pi ,t)=0, && 0<t; \end {aligned} \] where \(\alpha \) is a constant.

1. Determine the eigenfunctions of the homogeneous equation \[ u_{xx}=\lambda u,\qquad 0\le x\le \pi ,\qquad u_x(0)=u_x(\pi )=0. \]
2. Solve the inhomogeneous initial-Neumann problem by carrying out an eigenfunction expansion of \(u(x,t)\) in terms of the eigenfunctions obtained in part (a). That is, denoting the eigenfunctions by \(v_k\), integer \(k\), set \[ u(x,t)=\sum _{k\ge 0} c_k(t)v_k(x), \] and determine the time-dependent coefficients \(c_k(t)\).
3. Give a physical interpretation of the solution in the limit \(\alpha \to 0\).

Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]

1. Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac 12[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
2. Determine the solution for the initial data \(u(x,0)=1\) if \(\norm {x}<a\), \(u(x,0)=0\) if \(x>\norm {a}\); \(u_t(x,0)=0\).
3. Determine the solution for the initial data \(u(x,0)=0\); \(u_t(x,0)=1\) if \(\norm {x}<a\), \(u_t(x,0)=0\) if \(x>\norm {a}\).

Suppose that \(u(\mathbf {x})\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(\mathbf {x}\in \Omega \) and \(r>0\) is chosen such that \(B_r(\mathbf {x})\subset \Omega \) (ball of radius \(r\) centered at \(\mathbf {x}\)) then \[ u(\mathbf {x})=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(\mathbf {x})}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(\mathbf {x})\le \frac {n}{\omega _n r^n}\int _{B_r(\mathbf {x})}u(\mathbf {y})\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.