UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations August 18, 2014.

Let \(f(t,x)\in C^1(\mathbb {R}\times \mathbb {R}^n,\mathbb {R}^n)\) be a continuously differentiable function. Show that there is a \(\epsilon >0\) and a unique function \(y\in C^1([0,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem for \(0\le t\le \epsilon \) \[ \begin {cases} \dfrac {dx}{dt}=f(t,x),\\ x(0)=x_0. \end {cases} \]

Let \(A\) be an \(n\times n\) real matrix and let \(\varphi (t,x_0)\) denote the solution of the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] Suppose that \(k\) not necessarily distinct eigenvalues of \(A\) have non-positive real part and the rest have positive real part \[ \Re \lambda _1\le \cdots \le \Re \lambda _k\le 0 <\Re \lambda _{k+1}\le \cdots \le \Re \lambda _n. \] Show that there is a \(k\)-dimensional subspace \(E\subset \mathbb {R}^n\) such that

1. \(E\) is forward invariant: for every \(x_0\in E\) we have \(\varphi (t,x_0)\in E\) for all \(t\ge 0\);
2. If \(x_0\notin E\) then the distance from \(\varphi (t,x_0)\) to \(E\) tends to infinity as \(t\to \infty \).

Let \(p(t)>0\) be a smooth function that is periodic of least period \(T>0\). Show that every solution approaches zero as \(t\to \infty \). \[ x''+p(t)x'+x=0 \]

Consider Selkov’s model of glycolysis, the process whereby living cells break down sugar, where \(x\) and \(y\) are concentrations of ADP and F6P. Show that it has a nonconstant periodic solution. \[ \begin {aligned} \dot {x} &= -x+\frac {y}{10}+x^2y,\\ \dot {y} &= \frac {1}{2}-\frac {y}{10}-x^2y \end {aligned} \] [Hint: show the polygon with vertices \((0,0)\), \((0,5)\), \((\frac {1}{2},5)\) and \((\frac {11}{2},0)\) is forward invariant.]

Show that for small \(\norm {\epsilon }\), there is a \(2\pi \)-periodic solution near zero to the inhomogeneous equation \[ \ddot {y}+5y=\epsilon \sin ^3 t. \]

Suppose that \(\rho (x,t)\) is the number density of cars evolving according to the traffic model \[ \frac {\partial \rho }{\partial t}+\frac {\partial (\rho u)}{\partial x}=0. \] with \(u\) the car speed. Take \(u=1-\rho \). A queue is building up at a traffic light \(x=1\) so that, when the light turns green at \(t=0\), \[ \rho (x,0)= \begin {cases} 0, & x<0\text { and }x>1,\\ x, & 0<x<1. \end {cases} \]

1. Use the method of characteristics to solve the Cauchy problem. In particular, construct the characteristic projections and identify any shocks or rarefaction waves.
2. Hence, show that a collision first occurs at \(x=1/2\) when \(t=1/2\) and, thereafter, there is a shock \(S(t)\). Use the Rankine-Hugoniot condition to derive the following expression for the speed of the shock: \[ \frac {dS}{dt}=\frac {S+t-1}{2t}. \]

Consider the global Cauchy problem \[ u_t=Du_{xx}+bu_x+cu,\qquad x\in \mathbb {R},\ t>0,\qquad u(x,0)=g(x), \] where \(D>0\), \(b\) and \(c\) are constant coefficients.

1. Convert to a diffusion equation by setting \(v(x,t)=u(x,t)\exp (hx+kt)\) for appropriately chosen \(h,k\).
2. Solve the diffusion equation for \(v\) using a transform method.
3. Hence, show that if \(c<0\) and \(g\) is bounded, then \(u(x,t)\to 0\) as \(t\to \infty \).

A problem in the dynamics of the overhead power wire for an electric locomotive leads to the model \[ \frac {\partial ^2u}{\partial t^2} =\frac {\partial ^2u}{\partial x^2} \quad \text {for }x\ne X(t),\ t>0, \] where \(X\) is a prescribed smooth function with \(0<X'(t)<1\). Here \(X(t)\) is the locomotive position and \(u\) is the displacement of the wire. Across \(x=X\) there are prescribed discontinuities \[ \left [\frac {\partial u}{\partial x}\right ]_{X^-}^{X^+} =-V(X(t),t),\qquad [u]_{X^-}^{X^+}=0. \] Suppose that \(u=u_t=0\) at \(t=0\). By constructing a weak solution that takes into account the discontinuities across \(X\), show that \[ u(x,t)=\int _0^{\tau _0}V(X(\tau ),\tau )\left (1-X'(\tau )^2\right )d\tau , \] where the range of \(\tau \) is taken so that \(y=X(\tau )\) lies within the range \[ \tau -t<y-x<t-\tau ,\qquad \tau >0. \] [Hint: you will need to derive a relationship between discontinuities in \(\partial _tu\) and in \(\partial _xu\) along \(X(t)\). You will also need to consider the domain of dependence in the \((x,t)\)-plane].

A function \(u\in C^2(\Omega )\), \(\Omega \subset \mathbb {R}^n\), is subharmonic (superharmonic) in \(\Omega \) if \(\nabla ^2u\ge 0\) (\(\nabla ^2u\le 0\)) in \(\Omega \).

1. Show that if \(u\) is subharmonic then for every ball \(B_R(\mathbf {x})\Subset \mathbf {R}\) \[ u(\mathbf {x})\le \frac {1}{\omega _n R^{n-1}}\int _{\partial B_R(\mathbf {x})}u(\sigma )\dd \sigma . \] Hence show that \[ u(\mathbf {x})\le \frac {n}{\omega _n R^n}\int _{B_R(\mathbf {x})}u(\mathbf {y})\dd y. \] How are the inequalities changed if \(u\) is superharmonic?
2. Show that if \(u\in C(\overline {\Omega })\) is subharmonic (superharmonic), the maximum (minimum) of \(u\) is attained only on \(\partial \Omega \), unless \(u\) is constant.
3. Show that if \(u\) is harmonic in \(\Omega \) then \(u^2\) is subharmonic.
4. Let \(u\) be subharmonic in \(\Omega \) and \(F:\mathbb {R}\to \mathbb {R}\) smooth. Under what conditions on \(F\) is \(F(u)\) subharmonic?

Suppose \[ \nabla ^2u-cu=f\quad \text {in }D, \] with \[ u=g\quad \text {on }\partial D, \] where \(D\) is bounded and \(\partial D\) is smooth.

1. Assuming that a solution \(u\) exists, use an energy argument to show that it is unique when \(c>0\).
2. If \(c<0\) and \(f=g=0\), and \(D\) is the region \(r^2=x^2+y^2<1\), show that \[ u=a_0J_0(r\sqrt {-c}),\qquad a_0\text { constant}, \] is a solution as long as \(\sqrt {-c}\) is a zero of the Bessel function \(J_0(x)\), which satisfies \[ \frac {d^2J_0}{dx^2}+\frac {1}{x}\frac {dJ_0}{dx}+J_0=0. \]
3. If \(c<0\) and \(f=g=0\), and \(D\) is the region \(1<x^2+y^2+z^2<4\), show that there are non-trivial solutions if \(c=-n^2\pi ^2\), \(n=1,2,3\ldots \).