Problem 1 (qiuzhen-qualify 2023-fall)
Section: Analysis

(10 pts) For all $\xi \in \mathbb{R}$, compute
$$\int_{\mathbb{R}} \frac{e^{-ix\xi}}{(1 + x^2)^2}\,dx.$$

Problem 2 (qiuzhen-qualify 2023-fall)
Section: Analysis

(10 pts) Find all equilibrium points of the following system of ordinary differential equations for $(x(t), y(t))$ and determine their stability (stable or unstable):
$$\frac{dx}{dt} = \log(y^2 - x), \qquad \frac{dy}{dt} = x - y - 1.$$

Problem 3 (qiuzhen-qualify 2023-fall)
Section: Analysis

(10 pts) Find all distributions $u$ on $\mathbb{R}^2$ such that
$$x_1 x_2\,u = (x_1^2 - x_2^2)\,u = 0.$$

Problem 4 (qiuzhen-qualify 2023-fall)
Section: Analysis

(10 pts) Consider the initial value problem for Burgers' equation
$$\frac{\partial u}{\partial t} + u\,\frac{\partial u}{\partial x} = 0, \qquad (t,x) \in (0,\infty) \times \mathbb{R},$$
$$u(0,x) = -x + x^3, \qquad x \in \mathbb{R}.$$
Determine $T_{\max} > 0$ such that the problem has a $C^1$ solution in $(0, T_{\max}) \times \mathbb{R}$, but not in $(0, T) \times \mathbb{R}$ for any $T > T_{\max}$.

Problem 5 (qiuzhen-qualify 2023-fall)
Section: Analysis

(15 pts) Determine whether the following statement is true or not: If $f \in L^1(\mathbb{R})$ and $g \in L^1(\mathbb{R}) \cap L^\infty(\mathbb{R})$, then their convolution $f * g$ is a continuous function on $\mathbb{R}$ and satisfies
$$\lim_{|x|\to\infty} (f * g)(x) = 0.$$
Prove the statement if it is true and give a counterexample otherwise.

Problem 6 (qiuzhen-qualify 2023-fall)
Section: Analysis

(15 pts) Let $D = \{ z : |z| < 1 \} \subset \mathbb{C}$ be the unit disk and $g : D \to \mathbb{C}$ be a holomorphic function such that $g(0) = 0$ and $|\operatorname{{Re}} g(z)| < 1$ for any $z \in D$. Prove that for any $z \in D$,
$$|g(z)| \le \frac{2}{\pi} \log \frac{1 + |z|}{1 - |z|}.$$

Problem 7 (qiuzhen-qualify 2023-fall)
Section: Analysis

(15 pts) Let $\mathcal{H}$ be a complex Hilbert space, $T : \mathcal{H} \to \mathcal{H}$ be a bounded linear operator, $\sigma(T)$ be the spectrum of $T$. We also define the numerical range of $T$ to be
$$N(T) := \{\langle \psi, T\psi \rangle \mid \psi \in \mathcal{H}, \|\psi\| = 1\} \subset \mathbb{C}$$
where $(\cdot,\cdot)$ and $\|\cdot\|$ denote the inner product and the norm on $\mathcal{H}$, respectively.

(1) Show that $N(T)$ is a convex set in $\mathbb{C}$.

(2) For any $\psi \in \mathcal{H}$ and $z \in \mathbb{C}$, show that
$$\|(T - z)\psi\| \ge d(z, N(T))\,\|\psi\|.$$
Here $d(z, N(T))$ is the distance from $z$ to $N(T)$ in $\mathbb{C}$.

(3) Let $\overline{N(T)}$ be the closure of $N(T)$ in $\mathbb{C}$, show that $\sigma(T) \subset \overline{N(T)}$ and for any $z \in \mathbb{C} \setminus \overline{N(T)}$,
$$\|(T - z)^{-1}\| \le d(z, \overline{N(T)})^{-1}.$$
Here the norm on the left-hand side is the operator norm for bounded linear operator on $\mathcal{H}$.

Problem 8 (qiuzhen-qualify 2023-fall)
Section: Analysis

(15 pts) Consider the following initial value problem for the heat equation in $\mathbb{R}^{1+3}$:
$$\frac{\partial u}{\partial t} - \Delta u = N(u); \qquad t > 0,\ x \in \mathbb{R}^3,$$
$$u(0, x) = f(x) \in \mathcal{S}(\mathbb{R}^3).$$
Here $\mathcal{S}(\mathbb{R}^3)$ is the Schwartz class of functions on $\mathbb{R}^3$.

(1) If $N(u) = 0$, show that the unique solution $u = u(t, x)$ in $C([0, \infty); \mathcal{S}(\mathbb{R}^3))$ satisfies
$$\|u(t, \cdot)\|_{L^\infty(\mathbb{R}^3)} \le C(f)\,(1 + t)^{-3/2}$$
where $C(f)$ is a constant depending only on $f \in \mathcal{S}(\mathbb{R}^3)$.

(2) If $N(u) = u^4$, show that there exists $\varepsilon_0 > 0$ such that for any $f \in \mathcal{S}(\mathbb{R}^3)$ with
$$\sup_{\alpha,\beta \in \mathbb{N}^3,\ |\alpha| + |\beta| \le 1000} \ \sup_{x \in \mathbb{R}^3} |x^{\alpha} \partial^{\beta} f(x)| \le \varepsilon_0,$$
there is a solution $u = u(t, x) \in C^1\big((0, \infty); H^{10}(\mathbb{R}^3)\big).