Partial Differential Equations

Final Exam

1. Consider the IVP$u_x^2+u_y^2=1$,$u(x,y)=0$on the line$x+y=1$.

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

   2. [5 points] Solve the IVP

2. [5 points] Consider the PDE$-\Delta u = \lambda u$in$\Omega$,$u = 0$on$\partial\Omega$where$\lambda$is a scalar and$\Omega$is a bounded open set. If$\lambda \leq 0$, prove that$u \equiv 0$.

3. [5 points] Suppose$u$solves:

   $$\begin{align*} u_t - \Delta u & = u\text{ in } \Omega \times (0,T), \\ u(x,0) & = 0 \text{ in }\Omega, \\ u(x,t) & =0\text{ on }\partial\Omega\times[0,T]. \end{align*}$$

   Then show that$u(x,t) = 0$in$\Omega \times (0,T)$.

4. [5 points] Show that if$u$satisfies the heat equation$u_t - \Delta u = 0$in$\Omega \times (0,T)$, then the following maximum principle holds:

   $$\begin{equation*} \sup\limits_{\Omega \times (0,T)} u(x,t) = \sup\limits_{\Gamma_T} u(x,t). \end{equation*}$$

5. 1. [5 points] Prove the characteristic parallelogram property for one dimensional wave equations.

   2. [5 points] Use characteristic parallelogram property to solve

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

6. [5 points] Integrate the wave equation$u_{tt} - c^2u_{xx} = f(x,t)$in the characteristic triangle$P(x,t)$,$Q(x-ct,0)$,$R(x+ct,0)$to derive a formula for the solution.

7. [5 points] Is there an$f$in$L^1(\mathbb{R})$such that$f * f = f$and$f\neq 0$?

8. [5 points] Find a smooth function$a : \mathbb{R}^2 \to \mathbb{R}$such that, for the equation of the form

   $$a(x, y) \, u_x + u_y = 0,$$
   there does not exist any solution in the entire$\mathbb{R}^2$for any nonconstant initial value prescribed on$\{ y = 0 \}$.