UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations January 5th, 2017.

1. Find the first three successive approximations \(u_1(t)\), \(u_2(t)\), \(u_3(t)\) for the initial value problem (IVP) \[ \dot {x}=x^2,\qquad x(0)=1. \] Use mathematical induction to show that for all \(n\ge 1\), \(u_n(t)=1+t+\cdots +t^n+O(t^{n+1})\) as \(t\to 0\).
2. Solve the IVP in part (a) and show that the function \(x(t)=1/(1-t)\) is a solution to the IVP on the interval \((-\infty ,1)\). Compare with the approximation of part (a).
3. 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 \(X\). 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 \[ \norm {K(x)-K(y)}\le L_0T\norm {x-y}\qquad x,y\in V. \]

Consider the following linear equation for \(x\in \mathbb {R}^n\): \[ \dot {x}=A+B(t)x,\qquad x(t_0)=x_0. \] where \(A,B(t)\) are \(n\times n\) matrices. Suppose that all eigenvalues \(\lambda _j\), \(j=1,\ldots ,n\), of the matrix \(A\) satisfy \(\Reop (\lambda _j)<0\), and let \(B(t)\) be continuous for \(0\le t<\infty \) with \(\int _0^\infty \norm {B(t)}\dd t<\infty \).

1. Using the variation of constants formula show that there exist constants \(K,\sigma >0\) such that \[ \abs {x(t)} \le Ke^{-\sigma (t-t_0)}\abs {x_0} +K\int _{t_0}^t e^{-\sigma (t-s)}\abs {x(s)}\norm {B(s)}\dd s. \]
2. Let \(u(t)=e^{\sigma t}\abs {x(t)}\), \(v(t)=\norm {B(t)}\) and \(c=Ke^{\sigma t_0}\abs {x_0}\). Show that the inequality of part (a) can be rewritten as \[ u(t)\le c+\int _{t_0}^t v(s)u(s)\dd s. \]
3. From Gronwall’s inequality we have \[ u(t)\le c\exp \left (\int _{t_0}^t v(s)\dd s\right ). \] Use this to show that the zero solution of the IVP is asymptotically stable.

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 \(|\ldots |\) in the standard integral is crucial for this.

1. 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, and determine it in the case of a trajectory for which \(\abs {z(0)}<1\). Sketch the flow.
2. 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 \(A=\{x\in \mathbb {R}^2\mid 1<\abs {x}<\sqrt {2}\}\).

Does the partial differential equation \[ \frac {\partial u}{\partial t}-\frac {\partial u}{\partial x}=-u \] subject to the restriction that \(u=\exp (-x^2)\) on the curve \(t=x^2\) have a solution? If it does have a solution, indicate for what ranges of \(x\) and \(t\) this solution is uniquely defined and if it does not have a solution, explain why.

The Ornstein-Uhlenbeck (OU) differential equation is \[ \frac {\partial p}{\partial t} =\frac {\partial }{\partial x}(kxp)+D\frac {\partial ^2p}{\partial x^2}. \]

1. What type of partial differential equation is this? What calculation is needed to justify this statement?
2. If \(t\) has units of time and \(x\) has units of length, what are the units of the parameters \(k\) and \(D\)?
3. Express this equation in nondimensional variables? How many nondimensional parameters remain after nondimensionalization?
4. Find the solution of the OU equation on the domain \(-\infty <x<\infty \), \(t\ge 0\) with initial data \(p(x,0)=\delta (x-x_0)\) for arbitrary \(x_0\). Give an interpretation of this solution in terms of a probability.

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

Consider the partial differential equation defined for \(-\infty <x<\infty \) \[ \frac {\partial u}{\partial t} -\frac {\partial ^2u}{\partial x^2} =\chi (x)f(u), \qquad \chi (x)= \begin {cases} 0 & 0<x<L,\\ 1 & \text {elsewhere} \end {cases} \] with \(f(u)\) a bistable function, \(f(0)=f(a)=f(1)=0\), \(0<a<1\), \(f'(0)<0\), \(f'(1)<0\) and \[ \int _0^1 f(u)d u>0. \]

1. Prove that for all \(L>0\) sufficiently large, there is a standing, monotone increasing \(C^1\) profile with limiting behavior \(u(-\infty )=0\), \(u(\infty )=1\).
2. State a comparison theorem that can be used to show that if \(L\) is sufficiently large, the region \(0<x<L\) acts as a “blocking region” to propagation for this equation.

1. Suppose an initial profile \(u_0(x)\) is monotone with \(\lim _{x\to -\infty }u_0(x, )=u_L\), \(\lim _{x\to \infty }u_0(x)=u_R\). Under what conditions on \(u_L\) and \(u_R\) does the solution of Burger’s equation \[ \frac {\partial u}{\partial t} +u\frac {\partial u}{\partial x}=0, \] evolve into a shock, and what is its asymptotic speed?
2. Consider the viscous Burger’s equation \[ \frac {\partial u}{\partial t} +u\frac {\partial u}{\partial x} =\epsilon \frac {\partial ^2u}{\partial x^2}. \] with \(\epsilon >0\). Under what conditions on \(u_L\) and \(u_R\), where \(\lim _{x\to -\infty }u(x,t)=u_L\), \(\lim _{x\to \infty }u(x,t)=u_R\) does this equation have a travelling wave solution, and what is its speed? Verify your result.