UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations August 18, 2015.

Let \(f\in C^1(U,\mathbb {R}^n)\) for \(U\subset \mathbb {R}^n\) and \(x_0\in U\). Given the Banach space \(X=C([0,T],\mathbb {R}^n)\) with norm \(\norm {x}=\max _{0\le t\le T}\abs {x(t)}\), let \[ K(x)(t)=x_0+\int _0^t f(x(s))\dd s \] for \(x\in X\). Define \(V=\{x\in X\mid \norm {x-x_0}\le \epsilon \}\) for fixed \(\epsilon >0\) and suppose \(K(x)\in V\) (which holds for sufficiently small \(T\)), so that \(K:V\to V\) with \(V\) a closed subset of \(X\).

1. Give the definition of a locally Lipschitz function in an open set \(U\) of a normed vector space.
2. Using the fact that \(f\) is locally Lipschitz in \(U\) with Lipschitz constant \(L_0\), and taking \(x,y\in V\) show that \[ \abs {K(x(t))-K(y(t))}\le L_0t\norm {x-y}. \] Hence, show that \[ \norm {K(x)-K(y)}\le L_0T\norm {x-y},\qquad x,y\in V. \]
3. State the contraction mapping principle on a Banach space.
4. Choosing \(T<1/L_0\), apply the contraction mapping principle to show that the integral equation has a unique continuous solution \(x(t)\) for all \(t\in [0,T]\) and sufficiently small \(T\). Hence establish existence and uniqueness of the initial value problem \[ \frac {dx}{dt}=f(x),\qquad x(0)=x_0. \]

Consider the \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\qquad x\in \mathbb {R}^n,\qquad A(t)=A(t+T) \] Let \(\Phi (t)\) be a fundamental matrix with \(\Phi (0)=I\).

1. Show that there exists at least one nontrivial solution \(\chi (t)\) such that \[ \chi (t+T)=\mu \chi (t) \] where \(\mu \) is an eigenvalue of \(\Phi (T)\).
2. Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are then \(n\) linearly independent solutions of the form \[ x_i=p_i(t)e^{\rho _i t} \] where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho _i\) related to \(\mu _i\)?
3. Consider the equation \(\dot {x}=f(t)A_0x\), \(x\in \mathbb {R}^2\), with \(f(t)\) a scalar \(T\)-periodic function and \(A_0\) a constant matrix with real distinct eigenvalues. Determine the corresponding Floquet multipliers.

Consider the differential operator acting on \(L^2(\mathbb {R})\), \[ L=-\frac {d^2}{dx^2},\qquad 0\le x<\infty \] with self-adjoint boundary conditions \(\psi (0)/\psi '(0)=\tan \theta \) for some fixed angle \(\theta \).

1. Show that when \(\tan \theta <0\) there is a single negative eigenvalue with a normalizable eigenfunction \(\psi _0(x)\) localized near the origin, but none when \(\tan \theta >0\).
2. Show that there is a continuum of eigenvalues \(\lambda =k^2\) with eigenfunctions \(\psi _k(x)=\sin (kx+\eta (k))\), where the phase shift \(\eta \) is found from \[ e^{i\eta (k)}=\frac {1+ik\tan \theta }{\sqrt {1+k^2\tan ^2\theta }}. \]
3. Evaluate the integral \[ I(x,x')=\frac {2}{\pi }\int _0^\infty \sin (kx+\eta (k))\sin (kx'+\eta (k))\dd k, \] and interpret the result with regards the relationship to the Dirac Delta function and completeness, that is, \(\delta (x-x')-I(x,x')=\psi _0(x)\psi _0(x')\). You will need the following standard integral \[ \int _{-\infty }^{\infty } e^{ikx}\frac {1}{1+k^2t^2}\frac {\dd k}{2\pi } =\frac {1}{2\abs {t}}e^{-\abs {x/t}}. \]

HINT: you should monitor how the bound state contribution (for \(\tan \theta <0\)) switches on and off as \(\theta \) is varied. Keeping track of the modulus signs \(\abs {\cdots }\) in the standard integral is crucial for this.

Consider the nonlinear equation \[ \ddot {x}+\epsilon \dot {x}(x^2-1)+x=0,\qquad 0<\epsilon \ll 1 \] and choose initial conditions \(x(0)=1\), \(\dot {x}(0)=0\).

1. Using the method of multiple scales show that this has an asymptotic series solution of the form \[ x(t)\sim R(\epsilon t)\cos (t+\theta (\epsilon t))+\mathcal {O}(\epsilon ) \] with \[ \theta _\tau =0,\qquad R_\tau =\frac {1}{8}R(4-R^2) \] where \(\tau =\epsilon t\).
2. Derive the solution \[ R(\tau )=\frac {2}{(1+a_0e^{-\epsilon t})^{1/2}} \] and determine \(a_0\) from the initial conditions. Hence, establish that there exits a stable periodic orbit. [HINT: perform the change of variable \(y=R^2\).]

1. Show how to convert the ODE \[ \dot {x}=f_1(x,y),\qquad \dot {y}=f_2(x,y). \] into polar coordinates.
2. Consider the dynamical system \[ \begin {aligned} \dot {x} &= -y+x(1-z^2-x^2-y^2),\\ \dot {y} &= x+y(1-z^2-x^2-y^2),\\ \dot {z} &= 0. \end {aligned} \] Determine the invariant sets and attracting set of the system. Give a general definition of the \(\omega \)-limit set for a flow \(\phi (x,t)\) in \(\mathbb {R}^n\), and determine it in the case of a trajectory for which \(\abs {z(0)}<1\).
3. Use the Poincare-Bendixson (PB) Theorem and the fact that the planar system \[ \dot {x}=x-y-x^3,\qquad \dot {y}=x+y-y^3 \] has only the one critical point at the origin to show that this system has a periodic orbit in the annular region \(A=\{x\in \mathbb {R}^2\mid 1<\abs {x}<\sqrt {2}\}\).

Consider the problem of a thin layer of paint of thickness \(h(x,t)\) and speed \(u(x,y,t)\) flowing down a wall, see Fig. 1. The paint is assumed to be uniform in the \(z\)-direction. The balance between gravity and viscosity (fluid friction) means that the velocity satisfies the equation \[ \frac{\partial^2u}{\partial y^2}=-c, \] where \(c\) is a positive constant. This is supplemented by the boundary conditions \[ u(x,0,t)=0,\qquad \left.\frac{\partial u(x,y,t)}{\partial y}\right|_{y=h}=0. \] The density of paint per unit length in the \(x\)-direction is \(\rho_0h(x,t)\) where \(\rho_0\) is a constant, and the corresponding flux is \[ q(x,t)=\rho_0\int_0^h u(x,y,t)\,d y. \] **(a)** Using conservation of paint, and solving for \(u(x,y,t)\) in terms of \(h(x,t)\) and \(y\), derive the following PDE for the thickness \(h\): \[ \frac{\partial h}{\partial t}+ch^2\frac{\partial h}{\partial x}=0. \] **(b)** Set \(c=1\). Show that the characteristics are straight lines and that the Rankine-Hugoniot condition on a shock \(x=S(t)\) is \[ \frac{dS}{dt}=\frac{[h^3/3]_-^+}{[h]_-^+}. \] A stripe of paint is applied at \(t=0\) so that \[ h(x,0)= \begin{cases} 0, & x<0\text{ or }x>1,\\ 1, & 0 **Diagram:** See the accessible exam PDF linked above.

Suppose that \(u(x)\) 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 \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _n r^n}\int _{B_r(x)}u(y)\dd y. \]
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.

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.

1. Consider a string clamped at the end points \(a,b\) with \(u(a,t)=u_a\), \(u(b,t)=u_b\), where \(u(x,t)\) is the string’s deviation from the horizontal rest position. The kinetic energy of the string is \[ E_k=\frac {1}{2}\int _a^b \rho u_t^2\dd s,\qquad \dd s=\sqrt {1+u_x^2}\dd x, \] where \(\rho (x,t)\) is the mass density. The potential energy due to stretching of the string is \[ E_p=\int _a^b k\left (\sqrt {1+u_x^2}-1\right )\dd x, \] where \(k(x,t)\) is the elastic coefficient.

   1. Find the first variation of the action \[ A=\int _0^T(E_k-E_p)\dd t. \]
   2. Integrating by parts, imposing the boundary conditions, and equating the first variation to zero, derive the PDE \[ \left (\rho u_t\sqrt {1+u_x^2}\right )_t = \left (k(1+u_x^2)^{-1/2}u_x\right )_x. \] Assuming \(\rho ,k\) are constants and using the approximation \(\abs {u_x}\ll 1\) derive the classical one-dimensional wave equation.
2. Consider the functional \[ Y[u]=\int _D\left (\frac {1}{2}\abs {\nabla u}^2+\frac {1}{2}u^2+fu\right )\dd x, \] where \(D\) is a bounded domain in \(\mathbb {R}^n\) and \(f\) is a bounded, continuous function in \(D\). Suppose that there exists a minimizer \(u^*\) of \(Y[u]\) in the Sobolev space \(H_1(D)\). Use this to prove existence and uniqueness of the weak formulation of the PDE (for which \(u\in C^1(D)\)) \[ -\nabla ^2u+u=-f,\qquad x\in D\subset \mathbb {R}^n,\qquad \partial _nu=0,\qquad x\in \partial D. \] [Hint: show that the weak formulation can be expressed as \((u,\psi )_{H_1(D)}=0\) for all \(\psi \in H_1(D)\).]

Consider the one-dimensional parabolic PDE \[ \frac {\partial u}{\partial t} +\frac {\partial }{\partial x}\left (\frac {u^n}{n}\right ) =\epsilon \frac {\partial ^2u}{\partial x^2}. \]

1. Show that a traveling wave solution \(u(x,t)=U(z)\) with \(z=(x-Vt)/\epsilon \) satisfies \[ \frac {dU}{dz}=\frac {U^n}{n}-VU+\text {constant}, \] and deduce that \[ V=\frac {[U^n/n]_{-\infty }^{\infty }}{[U]_{-\infty }^{\infty }}. \]
2. Discuss how the traveling wave solution relates to shock solutions of the quasilinear equation obtained by setting \(\epsilon =0\). Also show that, when \(n=2\), \(U\) can only tend to \(U(\pm \infty )\) as \(z\to \pm \infty \) if \(dU/dz<0\).