UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations August 14, 2019.

1. Let \(f(t,x):\mathbb {R}\times \mathbb {R}^n\to \mathbb {R}^n\) be a continuously differentiable function. Show that there is an \(\epsilon >0\) and a unique continuously differentiable function \(y(t)\in \mathbb {R}^n\) for \(t\in [0,\epsilon ]\) that satisfies the initial value problem \[ \frac {dx}{dt}=f(t,x),\qquad x(0)=x_0. \]
2. Construct the first few Picard iterates for the differential equation \[ \dot {x}=2t-\sqrt {\max (0,x)},\qquad x(0)=0. \tag {1} \] Show that the Picard iterates do not converge. Why do they not converge?

Suppose \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\) and periodic of period \(T>0\). Consider the linear differential equation in \(\mathbb {R}^n\) \[ \begin {cases} \dfrac {dx}{dt}=A(t)x,\\ x(0)=x_0. \end {cases} \tag {2} \] Let \(\Phi (t)\) be the fundamental solution matrix with \(\Phi (0)=I\).

1. 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 \(\mu _i\)?
2. What is the fundamental solution matrix and what are the Floquet exponents for the system \[ A(t)= \begin {pmatrix} a & 1\\ 0 & \dfrac {\dot {p}(t)}{p(t)} \end {pmatrix}, \tag {3} \] where \(p(t)\) is a \(T\)-periodic function?

Let \(p(t)>\alpha >0\) be a smooth function defined for all \(t\). Prove that for any solution of \[ x''+p(t)x'+x=0, \tag {4} \] \(x(t)\to 0\) as \(t\to \infty \).

Under what conditions on the parameters \(a\) and \(b\) does the boundary value problem \[ u''+u=\sin (x),\qquad u(0)=a,\qquad u(\pi )=b, \tag {5} \] have a solution? Is the solution, if it exists, unique? If not, why not?

1. For a general dynamical system, give the definitions of an invariant set, attracting set and \(\omega \)-limit sets.
2. For the system \[ \dot {x}=x-y-x^3,\qquad \dot {y}=x+y-y^3, \] identify all critical points and their stability, and use the Poincare-Bendixson theorem to show that this system has a periodic orbit in the annular region \(1<x^2+y^2<2\).

Formulate and prove the maximum principle for Laplace’s equation \(u_{xx}+u_{yy}=0\).

Formulate and prove the maximum principle for the heat/diffusion equation \(u_t=u_{xx}\).

Show that the heat/diffusion equation with the wrong sign \(u_t=-u_{xx}\) is extremely unstable (and so, the Cauchy problem for this equation is ill posed in the sense of Hadamard).

Explain the fundamental difference of wave equations in 1D, 2D, and 3D: In 3D — unlike 1D and 2D — any signal (with finite support) propagates leaving no trace behind it.

Show that a real symmetric linear operator has a full set of real orthogonal eigenvectors (even if it has multiple eigenvalues). [Assume that the operator is a matrix acting in \(\mathbb {R}^n\).]