UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph. D. Preliminary Examination in Differential Equations August 13, 2012.

Consider Griffith’s model for a genetic control system, where \(x\) and \(y\) are proportional to concentration of protein and the messenger RNA from which it is translated, respectively, and \(\mu >0\) is a rate constant \[ \begin {aligned} \dot {x} &= y-\mu x,\\ \dot {y} &= \frac {x^2}{1+x^2}-y. \end {aligned} \]

1. Show that the system has three fixed points when \(\mu <\mu _c\) and one when \(\mu >\mu _c\) where \(\mu _c\) is to be determined.
2. What is the nature of the bifurcation at \(\mu =\mu _c\)?

Consider a Brusselator system for chemical reactants \(x,y\) given by \[ \begin {aligned} \dot {x} &= 1-4x+x^2y,\\ \dot {y} &= 3x-x^2y \end {aligned} \]

1. Show that the quadrilateral bounded by the lines \(x=0\), \(y=0\), \(y=8+x\) and \(y=10-x\) is a forward invariant subset. [Hint: need to show that if \(n\) is the outer normal to a given edge of the quadrilateral, then \(n\cdot (\dot {x},\dot {y})\le 0\).]
2. Find the fixed points and determine their stability.
3. Show that the system has at least one nonconstant periodic solution.

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Construct an approximation to the center manifold at the origin and use it to determine the local behavior of solutions. \[ \begin {aligned} \dot {x} &= -xy,\\ \dot {y} &= -y+x^2-2y^2. \end {aligned} \]

Consider the second-order equation \[ \nabla ^2u-cu=f\qquad \text {in }D\subset \mathbb {R}^n \] with \(c\) constant, \(u,f\in C^2(D)\) and \[ u=g\qquad \text {on }\partial D, \] where \(D\) is bounded and \(\partial D\) is smooth.

1. Assuming a solution \(u\) exists, show that it is unique when \(c>0\). Establish this using two different methods: (i) an integral method and (ii) a Maximum Principle.
2. Suppose that \(n=2\), \(c<0\), \(f=g=0\), and \(D\) is the region \(r^2=x^2+y^2<1\). Show that for \(a_0\) constant, \[ u=a_0J_0(r\sqrt {-c}), \] is a solution as long as \(\sqrt {-c}\) is a zero of the Bessel function \(J_0(x)\), which satisfies \[ \frac {d^2J_0}{dx^2}+\frac 1x\frac {dJ_0}{dx}+J_0=0. \]
3. Suppose that \(n=3\), \(c<0\), \(f=g=0\), and \(D\) is the region \(1<x^2+y^2+z^2<4\). Show that there are non-trivial solutions if \(c=-n^2\pi ^2\), \(n=1,2,3\ldots \).

Suppose that \(\rho (x,t)\) is the number density of cars per unit length along a road, \(x\) being distance along the road, such that \[ \frac {\partial \rho }{\partial t} +\frac {\partial [\rho (1-\rho )]}{\partial x}=0. \]

1. Show that \(\rho \) is constant along the characteristics \[ \frac {dx}{dt}=1-2\rho , \] and derive the following Rankine-Hugoniot condition for the speed of a shock \(x=S(t)\): \[ \frac {dS}{dt}=\frac {[\rho (1-\rho )]_-^+}{[\rho ]_-^+}. \]
2. A queue is building up at a traffic light \(x=1\) so that, when the light turns to green at \(t=0\), \[ \rho (x,0)= \begin {cases} 0, & \text {if }x<0\text { and }x>1,\\ x, & \text {if }0<x<1. \end {cases} \] Solve the corresponding characteristic equations, and sketch the resulting characteristic curves. Deduce that a collision first occurs at \(x=1/2\) when \(t=1/2\), and that thereafter there is a shock such that \[ \frac {dS}{dt}=\frac {S+t-1}{2t}. \]

According to Maxwell’s Equations, the static magnetic field \(H=(H_1,H_2,H_3)^T\) in an inhomogeneous medium satisfies \[ \nabla \cdot (\mu H)=0,\qquad \nabla \times H=0, \] where \(\mu \) is a space-dependent permeability. Suppose the interface \(z=0\) separates a medium in which \(\mu =\mu _+\) from one in which \(\mu =\mu _-\) with \(\mu _\pm \) constants.

1. Define the weak solution: for every given simply connected, smooth domain \(\Omega _3\subset \mathbb {R}^3\) and arbitrary embedded smooth two dimensional surface with boundary \(\Omega _2\subset \mathbb {R}^3\), assuming enough regularity, integrate by parts the equations \[ \int _{\Omega _3}\psi \nabla \cdot (\mu H)\dd V=0,\qquad \int _{\Omega _2}\psi \nabla \times H\cdot \dd S=0, \] where \(\psi \) is an arbitrary, suitable test function.
2. Taking \(\Omega _3\) to be a cylindrical domain that cuts the interface vertically, use Green’s Theorem to show that \[ [\mu H_3]_-^+=0. \]
3. Taking \(\Omega _2\) to be a rectangular surface that cuts the interface vertically, use Stokes’ Theorem to show that \[ [H_1]_-^+=0=[H_2]_-^+. \]

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

1. The total energy \(E=k(t)+p(t)\) is constant in \(t\).
2. Assume that there are two distinct solutions \(u,v\) to the initial value problem. By considering the energy of the difference solution \(U=u-v\), prove that \(u=v\), that is, the solution is unique.

1. For a smooth domain \(D\subset \mathbb {R}^n\), solve Poisson’s equation \[ \nabla ^2u=f\text { in }D,\qquad \frac {\partial u}{\partial n}=g\text { on }\partial D \] in terms of an appropriately defined Green’s function.
2. Show that the two-dimensional Green’s function for the Laplacian in an unbounded domain is \[ G(x,x')=\frac {1}{2\pi }\ln \lvert x-x'\rvert . \]
3. Use the method of images to derive the Green’s function for the Laplacian in the half-space \(x\in \mathbb {R}\), \(y>0\) with a Neumann boundary condition on \(y=0\), and evaluate the corresponding solution of part (a) when \(f=0\).