Partial Differential Equations

Midterm

1. [2 points] Reduce to canonical form :$u_{xx}-2u_{xy}-3u_{yy}+u_y=0$.

2. Evaluate the integrals:

   1. [1 point]$\displaystyle \int_\Omega\frac{2x_1}{1+|x|^2}~dx$, where$\Omega=\{x\in \mathbb{R}^3:|x_1|+|x_2|+|x_3|\leq 1\}$

   2. [1 point]$\displaystyle \int_{B(\alpha,1)}\frac{\partial u}{\partial x_1}dx$, where$u=|x|^{-1}$in$\mathbb{R}^3$and$\alpha=(2,0,0)$.

3. Consider the PDE$xu_x+yu_y+zu_z=3u$in$\mathbb{R}^3$.

   1. [3 points] Solve the PDE with initial condition$u(x,y,1)=x^2+y^2$.

   2. [1 point] Is it possible to find unique solution if the initial condition is prescribed on the surface$z=1+x^2+y^2$?

4. Consider the following IVPs:

   • A:$u=u_x^2-3u_y^2,\ u(x,0)=x^2,\ x>0$.

   • B:$u=u_xu_y,\ u(x,0)=x^2,\ x>0$.

   1. [4 points] Discuss the existence and uniqueness of both IVPs.

   2. [3 points] Solve any one the above.

5. [3 points] Let$\Omega$be an open, bounded set in$\mathbb{R}^n$. Suppose$u \in C^2(\Omega) \cap C(\bar{\Omega})$satisfies$\Delta u = -1$in$\Omega$,$u = 0$on$\partial\Omega$. Show that for$x \in \Omega$,$u(x) \geq \frac{1}{2n}(d(x, \partial\Omega))^2$.

   (Hint: For fixed$x_0 \in \Omega$, consider the function$u(x) + \frac{1}{2n}|x - x_0|^2$,$x \in \Omega$.)

6. [3 points] Suppose$u$is a harmonic function in$\mathbb{R}^n$satisfying$|u(x)| \leq C(1 + |x|^m)$, for some non-negative integer$m$and for all$x \in \mathbb{R}^n$. Show that$u$is a polynomial of degree at most$m$.

7. [3 points] Let$\Omega$is a bounded, open subset of$\mathbb{R}^n$, and$u\in C^1(\Omega)$. If$\int_{\partial B}\frac{\partial u}{\partial \nu}dS=0$for every ball$B$with$\bar{B}\subset\Omega$, show that$u$is harmonic in$\Omega$.

8. Consider the PDE$xu_x+yu_y=2u$on$\mathbb{R}^2$.

   1. [3 points] Solve the PDE with the initial condition$u(x,1)=x$. Determine whether the solution is globally unique? If it is not, find ans alternative solution on$\mathbb{R}^2$.

   2. [3 points] Find two solutions to the PDE with the initial condition$u(x,e^x)=xe^x$, ensuring that these solutions do not coincide in any neighborhood of the initial curve.