UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations January 4, 2014.

Let \(f(x)\) be continuously differentiable for all \(X\in \mathbb {R}^n\) and let \(x_0\in \mathbb {R}^n\times \mathbb {R}\). 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 \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0. \end {cases} \]

Let \(A\) be an \(n\times n\) real matrix such that \(\Re \lambda <0\) for all eigenvalues \(\lambda \) of \(A\). Show that there are positive constants \(c\) and \(\mu \) such that any solution \(x(t)\) of \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] satisfies \[ \norm {x(t)}\le ce^{-\mu t}\norm {x_0}\qquad \text {for all }t\ge 0. \]

Show that for small \(\norm {\epsilon }\), there is a \(2\pi \)-periodic solution to the forced Duffing’s equation \[ \ddot {y}+3y+\epsilon y^3=\cos t. \]

Show that the equation has a non constant periodic solution. \[ \ddot {x}+(x^2+\dot {x}^2-1)\dot {x}+2x=0 \]

Consider the system \[ \begin {aligned} \dot {x} &= \mu +x^2+y^2,\\ \dot {y} &= -y+x^2 \end {aligned} \]

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 at the origin at \(\mu =0\).

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.