UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations January 3, 2020.

The goal of this problem is to use Picard iterates to find the solution of the initial value problem \[ \frac {dx}{dt}=x^2,\qquad x(0)=1 \tag {1} \]

1. State a theorem, without proof, with necessary conditions on \(f(x,t)\), guaranteeing the existence of a unique solution of the differential equation \(\frac {dx}{dt}=f(x,t)\), \(x(0)=x_0\).
2. Verify that the function \(f=x^2\) satisfies these necessary conditions.
3. Define the Picard iterates for the equation (1) and calculate the first three functions \(x_0(t)\), \(x_1(t)\), and \(x_2(t)\).
4. Prove by induction that \[ x_n(t)=1+t+t^2+\cdots +t^n+O(t^{n+1}) =\sum _{j=0}^{n}t^j+O(t^{n+1}). \tag {2} \] Hint: You will need to show that \[ \sum _{j=0}^{n}t^j\sum _{k=0}^{n}t^k =\sum _{j=0}^{n}jt^j+O(t^{n+1}). \]
5. Find the exact solution of (1). How does it compare to the approximate solution found in the previous part?

Consider the first order differential equation \[ \frac {dx}{dt}=f(x,t), \tag {3} \] where \(f(x,t)\) is periodic in \(t\), \(f(x,t+1)=f(x,t)\) for all \(x\) and \(t\).

1. Define the Poincare map for this differential equation.
2. Suppose \(f(x,t)=x(A(t)-x)\), and \(0<\alpha <A(t)<\beta \). Prove that the differential equation has at least one nontrivial periodic solution.

1. For a system of equations \(\frac {dx}{dt}=A(t)x\), where \(x\in \mathbb {R}^n\) and \(A(t)\) is a real \(n\times n\) matrix function which is smooth in \(t\), what is Abel’s formula? (State without proof).
2. For the linear differential equation \[ \frac {d^2x}{dt^2}-p(t)\frac {dx}{dt}+q(t)x=0 \tag {4} \] where \(p(t)>\alpha >0\) is a smooth function defined for all \(t\), prove there is at least one solution \(x(t)\) that is unbounded as \(t\to \infty \).

Under what conditions on the parameters \(a\) and \(b\) does the boundary value problem \[ u''+u=\cos (x),\qquad u'(0)=a,\qquad u'(\pi )=b \tag {5} \] have a solution?

1. For a general dynamical system, give the definitions of an invariant set, attracting set and \(\omega \)-limit sets.
2. For the system \[ \dot {x}=-f(x,y)x-g(x,y)y,\qquad \dot {y}=g(x,y)x-f(x,y)y, \tag {6} \] where \(g(x,y)>0\) for all \(x,y\), and \(f(x,y)=x^2+y^2+2y-1\), identify all critical points and their stability, and show that this system has an invariant annular region which encloses the origin, and therefore has a nontrivial periodic solution.

1. Show that the wave equation \(u_{tt}=c^2u_{xx}\) \((-\infty <x<\infty ,\ t>0)\) conserves energy \[ E=\int _{-\infty }^{\infty }\left [(u_t)^2+c^2(u_x)^2\right ]d x. \]
2. Show the uniqueness: If two solutions of the wave equation satisfy the same initial conditions, then the solutions are identical.

Obtain the following symmetry properties for the Cauchy problem for the 1D heat equation \[ u_t=ku_{xx}+f(x,t)\quad (-\infty <x<\infty ,\ t>0),\qquad u(x,0)=\phi (x),\qquad u_t(x,0)=\psi (x). \]

1. If the data is even in \(x\) [\(\phi (x)=\phi (-x)\), \(\psi (x)=\psi (-x)\), \(f(x,t)=f(-x,t)\)], then the solution is also even in \(x\) [\(u(x,t)=u(-x,t)\)].
2. If the data is odd in \(x\) [\(\phi (x)=-\phi (-x)\), \(\psi (x)=-\psi (-x)\), \(f(x,t)=-f(-x,t)\)], then the solution is also odd in \(x\) [\(u(x,t)=-u(-x,t)\)].
3. If the data is periodic in \(x\) [\(\phi (x+A)=\phi (x)\), \(\psi (x+A)=\psi (x)\), \(f(x+A,t)=f(x,t)\)], then the solution is also periodic in \(x\) with the same period \(A\) [\(u(x+A,t)=u(x,t)\)].

1. Why does the heat/diffusion equation imply infinite propagation speed?
2. Why does the wave equation imply finite propagation speed?

Suppose someone treats one of the variables in the Laplace equation as time and tries to solve the evolution problem \[ \begin {aligned} u_{tt}+u_{xx}&=0\quad (0<x<l,\ t>0),\\ u(0,t)&=u(l,t)=0,\qquad u(x,0)=\phi (x),\qquad u_t(x,0)=\psi (x), \end {aligned} \] similar to the wave equation. Show that this problem has no continuous dependence on data, even if the time \(t\) belongs to a finite interval \(0<t<T\) (\(T\) is a positive constant).

1. Show the uniqueness of Robin’s problem \[ \begin {aligned} \Delta u&=f(x)\ [x\in D],\\ \nabla u\cdot \mathbf {n}+a(x)u&=h(x)\ [x\in \partial D],\qquad [D\text { is a 3D domain, }a(x)>0]. \end {aligned} \]
2. When \(a\equiv 0\), we have the Neumann problem. Show that it has no solution, unless \[ \int _D fd x=\oint _{\partial D}hd A. \] Describe physical meaning of this fact in the case of steady heat conduction.
3. Prove the uniqueness of the Neumann problem up to an additive constant.