Partial Differential Equations

Supplementary Exam

1. Consider the IVP$u_y=u_x^3$,$u(x,0)=2x^\frac{3}{2}$.

   1. [5 points] Discuss the existence and uniqueness of the IVP.

   2. [5 points] Solve the IVP.

2. [10 points] Let$\Omega$be a domain in$\mathbb{R}^2$symmetric about the$x$-axis and let$\Omega^+ = \{(x, y) : y > 0\}$be the upper part. Assume$u \in C(\overline{\Omega^+})$is harmonic in$\Omega^+$with$u = 0$on$\partial\Omega^+ \cap \{y = 0\}$. Define

   $$v(x, y) = \begin{cases} u(x, y), & y \geq 0, (x, y) \in \Omega, \\ -u(x, -y), & y < 0, (x, y) \in \Omega. \end{cases}$$

   Show that$v$is harmonic.

3. Let$\phi$be fundamental solution of heat equation given by:

   $$\begin{equation*} \phi(x,t) = \begin{cases} \frac{1}{(4\pi t)^{n/2}} e^{-\frac{|x|^2}{4t}}, & x \in \mathbb{R}^n, t > 0, \\ 0, & x \in \mathbb{R}^n, t = 0. \end{cases} \end{equation*}$$

   1. [5 points] Show that

      $$\lim\limits_{(x,t)\to(x_0,0)} \phi(x,t) = 0, \quad \text{for } x_0 \neq 0.$$

   2. [5 points] Show that

      $$\int_{\mathbb{R}^n} \phi(x,t)dx = 1 \quad \forall t > 0.$$

4. [10 points] Solve

   $$\begin{align*} & u_{tt} - u_{xx} = 0,\quad x>0,\quad t>0, \\ & u(x,0) = f(x),\quad u_t(x,0) = g(x), \\ & u_x(0,t) = h(t). \end{align*}$$

5. [10 points] Consider the wave equation:

   $$u_{tt} - \Delta u = 0, \quad \text{in } \mathbb{R}^n \times (0,\infty),$$
   with initial data:
   $$u(x,0) = \phi(x), \quad u_t(x,0) = \psi(x).$$
   Verify that the solution is given by:
   $$u(x,t) = u_{\phi}(x,t) + \int_0^t u_{\psi}(x,s)ds,$$
   and also by:
   $$u(x,t) = v_{\psi}(x,t) + \frac{\partial}{\partial t} v_{\phi}(x,t).$$