UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph. D. Preliminary Examination in Differential Equations January 5, 2011.

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

Suppose that \(H:\mathbb {R}^2\to \mathbb {R}\) is a smooth function. Consider the Hamiltonian System \[ \begin {aligned} \dot {x} &= H_y(x,y),\\ \dot {y} &= -H_x(x,y). \end {aligned} \] Suppose that \(\Gamma _0\) is a nonconstant periodic orbit for this system. Show that \(\Gamma _0\) cannot be Asymptotically Orbitally Stable. (This is sometimes referred to as asymptotic stability of the periodic orbit \(\Gamma _0\) or asymptotic Poincarè stability.)

Determine whether the planar system \[ \begin {aligned} \dot {x} &= x-x^3-xy^2,\\ \dot {y} &= 2y-y^5-yx^4 \end {aligned} \] has any nonconstant periodic solutions.

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. \[ \begin {aligned} \dot {x} &= -y-xz,\\ \dot {y} &= x-y^3,\\ \dot {z} &= -z-2xy-2x^4+x^2. \end {aligned} \]

Suppose that \(u\in C^2\) solves \[ u_{tt}-\operatorname {div}(aDu)=0,\qquad \text {in }\mathbb {R}^n\times (0,\infty ), \] where \(a\in C^1(\mathbb {R}^n)\) satisfies \(\frac 12\le a(x)\le 1\) for all \(x\).

1. Assunimg \(\operatorname {supp}_x u(x,t)\subset B(0,R)\subset \mathbb {R}^n\) for all \(0\le t\le T\), show that the energy \[ E(t)=\frac 12\int _{\mathbb {R}^n}u_t^2+a\lvert Du\rvert ^2\dd x,\qquad 0\le t\le T, \] is conserved.
2. Using an appropriately defined local energy, establish that solutions exhibit finite propagation speed: if for some \(r>0\), \(u(x,0)\) and \(u_t(x,0)\) are zero for \(x\in B(x_0,r)\subset \mathbb {R}^n\), then \(u\) is zero in the cone \(K=\{(x,t):0\le t\le r,\lvert x-x_0\rvert <r-t\}\).

The time-dependent Schrödinger equation can be written \[ i u_t=-\Delta u+Vu, \] where \(u(x,t)\), \(x\in \mathbb {R}^n\), \(t\in \mathbb {R}\), is the complex-valued “wave function,” and \(V(x)\ge 0\) is a given continuous real-valued potential function.

1. Using complex exponential plane waves, describe the dispersion relation for this equation in case \(V\equiv 1\). Is this problem dispersive?
2. Assuming \(V\) is bounded, and the problem is formulated over the smooth bounded connected spatial domain \(U\subset \mathbb {R}^n\), with homogeneous Dirichlet boundary conditions, prove that the solutions to the time dependent problem can be expressed as a discrete sum involving solutions to the time-independent problem \[ E_k\phi _k=\Delta \phi _k+V\phi _k,\qquad \text {in }U \] where \(0<E_1\le E_2\le \cdots \).

Consider Burger’s equation \[ \begin {aligned} u_t+\left (\frac {u^2}{2}\right )_x &= 0, && \text {in }\mathbb {R}\times (0,\infty ),\\ u &= g, && \text {on }\mathbb {R}\times \{0\}. \end {aligned} \]

1. Construct an integral solution satisfying the Rankine-Hugoniot condition which exhibits “nonphysical shocks.”
2. Write down the explicit entropy condition for this problem and explain why it rules out the solution in part (a).
3. Construct another integral solution which satisfies the entropy condition, and explain why this condition does not allow shocks.

Let \(A(x)\) be an \(n\times n\) matrix-valued function defined on the smooth bounded connected domain \(U\subset \mathbb {R}^n\). Assume \(A(x)=(a_{i,j}(x))\) is uniformly symmetric positive definite, and \(a_{i,j}\in C^\infty (\overline {U})\) for all \(1\le i,j\le n\). Consider the problem \[ \begin {aligned} -\operatorname {div}(ADu) &= \lambda u, && \text {in }U,\\ u &= 0, && \text {on }\partial U. \end {aligned} \]

1. Characterize the principal (smallest) eigenvalue \(\lambda _1\) in terms of the Rayleigh quotient for this problem.
2. Prove that \(\lambda _1>0\), and that the associated eigenfunction \(u_1\) is a minimizer of the Rayleigh quotient.

Consider the problem \[ \begin {aligned} u_t &= u_{xx}, && t\in (0,1),\ x\in (0,1),\\ u(0,t)=u(1,t) &= 0, && t\in (0,1),\\ u(x,1) &= f, && x\in (0,1). \end {aligned} \]

1. What are the least restrictive condition you can find on \(f\) which guarantee that the problem admits a solution \(u\), bounded in the norm \[ \norm {u}_B=\sup _{t\in [0,1]}\left (\int _0^1 u(x,t)^2d x\right )^{1/2}\ ? \]
2. Under what conditions on \(f\) does the stability estimate \[ \norm {u}_B\le C\norm {f}_{L^2(0,1)} \] hold?
3. Assuming \(u\in C^2(\overline {U_1})\), where \(U_1=(0,1)\times (0,1]\), is uniqueness of solutions guaranteed?