UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations August 16, 2018.

Let \(f:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuous function and \(x_0\in \mathbb {R}^d\). Show that there is \(T>0\) such that the initial value problem \[ \begin {aligned} \dot {x}&=f(t,x)\\ x(0)&=x_0 \end {aligned} \] has a solution \(x(t)\in C^1([0,T),\mathbb {R}^n)\). Is \(x(t)\) unique? [Don’t just quote a result. State the theorem and give as complete and detailed a proof as you can.]

1. Let \(f:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuously differentiable function and \(x_0\in \mathbb {R}^d\) be such that \(f(t,x_0)=0\) for all \(t\ge 0\). Define: the constant solution \(x(t)=x_0\) on \(t\ge 0\) is a stable (Liapunov stable) solution for of \[ \dot {x}=f(t,x). \]
2. Let \(A\) be an \(d\times d\) real matrix. State and prove a theorem that gives necessary and sufficient conditions on \(A\) so that the the zero solution \(z(t)=0\) on \(t\ge 0\) is stable for \[ \dot {x}=Ax. \]
3. Let \(A(t)\) be a smooth, \(T\)-periodic for some \(T>0\), \(d\times d\) real matrix function. State but don’t prove a theorem that gives necessary and sufficient conditions on \(A(t)\) so that the the zero solution \(z(t)=0\) is stable for \[ \dot {x}=A(t)x. \]

Let \(g:[0,\infty )\times \mathbb {R}^d\to \mathbb {R}^d\) be a continuously differentiable function and \(p>1\) a constant such that \[ \abs {g(t,x)}\le \abs {x}^p\quad \text {for all }t\ge 0\text { and }x\in \mathbb {R}^d. \] Let \(A\) be a real matrix such that \(\operatorname {Re}\lambda <0\) for all eigenvalues of \(A\).

1. Show that there is \(\delta >0\) such that if \(\abs {x_0}\le \delta \), then the initial value problem \[ \begin {aligned} \dot {x}&=Ax+g(t,x)\\ x(0)&=x_0 \end {aligned} \tag {1} \] has a bounded solution on \([0,\infty )\). [Provide the complete proof. Do not just quote a theorem.]

2. Is the zero solution of (1) asymptotically stable? Explain.

   [Hint: let \(f(t)\), \(\varphi (t)\) be nonnegative continuous functions on the interval \(J=(\alpha ,\beta )\) containing \(t_0\). Let \(c_0\ge 0\). Gronwall’s Lemma says that if \[ f(t)\le c_0+\left |\int _{t_0}^{t}\varphi (s)f(s)\dd s\right | \] for all \(t\in J\) then \[ f(t)\le c_0\exp \left |\int _{t_0}^{t}\varphi (s)\dd s\right | \] for all \(t\in J\).]

Consider the torqued pendulum with constant drag \(\beta >0\) and torque \(\mu >1\) large enough to overcome the pull of gravity, no matter what the angle \(x\) of the pendulum may be. \[ \begin {aligned} \dot {x}&=y\\ \dot {y}&=\mu -\beta y-\sin x \end {aligned} \]

1. Show that the solution of the initial value problem with \(x(0)=0\) and \(y(0)=y_0\) exists for all time.
2. Show that there is a solution such that the pendulum continues to rotate indefinitely in a \(T\)-periodic fashion. It would have \(x^*(T+t)=x^*(t)+2\pi \) and \(y^*(T+t)=y^*(t)\).
3. Show that the periodic solution is unique.
4. Show that the periodic solution is orbitally stable.

Consider the bathtub model for the concentrations \(x(t)\ge 0\) and \(y(t)\ge 0\) \[ \begin {aligned} \dot {x}&=-9x+y+9x^2y\\ \dot {y}&=6-y-9x^2y. \end {aligned} \]

1. Give as complete a proof as you can that there exists a nonconstant periodic orbit.
2. State the Hartman-Grobman Theorem. What role does it play in your proof of (a)?

Consider Newton’s equation \[ \ddot {x}+\frac {x}{2+x^2}=\epsilon \cos t \] Prove that there is an \(\epsilon _0>0\) such that if \(\abs {\epsilon }<\epsilon _0\) then there is a \(2\pi \)-periodic solution.

Suppose that the initial value problem for the wave equation \[ \begin {aligned} \frac {\partial ^2u}{\partial t^2}-c^2\nabla ^2u&=0, &&x\in \mathbb {R}^n,\quad t>0,\\ u(0,x)&=g(x),\\ \frac {\partial u}{\partial t}(0,x)&=0, &&x\in \mathbb {R}^n \end {aligned} \] has a solution \(u_g\). Find (and verify) the solution \(u_h\) of the same equation subject to the initial conditions \[ \begin {aligned} u(0,x)&=g(x),\\ \frac {\partial u}{\partial t}(0,x)&=h(x), &&x\in \mathbb {R}^n \end {aligned} \] in terms of \(u_g\), \(g\) and \(h\).

Find and sketch the solution of the differential equation \[ u_t+uu_x=0 \] on the domain \(-\infty <x<\infty \), \(t>0\) subject to the initial condition \(u(0,x)=u_0(x)\), where

1. \[ u_0(x)= \begin {cases} 0; & \text {if }x<0,\\ x, & \text {if }0<x<1,\\ 1, & \text {if }x>1; \end {cases} \]
2. \[ u_0(x)= \begin {cases} 1; & \text {if }x<0,\\ 1-x, & \text {if }0<x<1,\\ 0, & \text {if }x>1. \end {cases} \]

For which (if any) of these initial profiles does the solution of the problem \[ u_t+uu_x=\epsilon u_{xx}, \] \(\epsilon >0\), converges to a traveling wave profile? What is the speed of the travelling wave solution?

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 \(u_R(x,t)=Re^{i(t+x/c_1)}\) and a transmitted wave \(u_T=Te^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients \(R,T\), and interpret the result.

Suppose that \(u(x)\in C^2(D)\) is a harmonic function on the smooth bounded domain \(D\subset \mathbb {R}^n\) so that \(\nabla ^2u=0\) in \(D\).

1. prove the mean value property: if \(x\in D\) and \(r>0\) is chosen so that the ball of radius \(r\) and center \(x\) satisfies \(B_r(x)\subset D\) then \[ u(x)=\frac {1}{\omega _nr^{n-1}}\int _{\partial B_r(x)}u(y)\dd y \] where \(\omega _n\) is the \(n-1\) dimensional area of \(\partial B_1\). Hence show that \[ u(x)=\frac {n}{\omega _nr^n}\int _{B_r(x)}u(z)\dd z. \]
2. Assuming that \(D\) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in D\) only if \(u\) is constant.

Consider the global Cauchy problem \[ u_y=Du_{xx}+bu_x+cu,\quad x\in \mathbb {R},\ t>0;\qquad u(0,x)=g(x), \] where \(b\), \(c\) \(D\) are constants such that \(D>0\).

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

Let \(D\subset \mathbb {R}^n\) be a bounded, connected domain with smooth boundary. Let \(\phi \) be an eigenfunction for the Laplacian on \(D\) with Dirichlet Boundary conditions (\(\phi =0\) on \(\partial D\)) and with eigenvalue \(\lambda \) \[ \Delta u+\lambda u=0. \tag {2} \]

1. Show that there is a finite constant \(C\) so that every nonzero continuously differentiable function \(u\in C^1(D)\) such that \(u=0\) on \(\partial D\) satisfies \[ \int _D u^2 \le C\int _D \abs {\nabla u}^2. \]
2. By integrating eigenfunctions of (2), conclude that eigenvalues are positive \(\lambda >0\).
3. Denote the least value of the Rayleigh Quotient \[ \lambda _1(D)= \inf _{\substack {u\in C_0^1(D)\\ u\not \equiv 0}} \frac {\int _D \abs {\nabla u}^2}{\int _D u^2}. \] which satisfies \(\lambda \ge 1/C\) by (a). 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 \(\lambda _1(D_1)\ge \lambda _1(D_2)\).