Problem set 3, PDE

Problem 1

Find solutions of the following initial-value problems in $latex \R^2$.

1. $latex \partial_y u + (1+x^2)\partial_x u - u = 0; u(x,0) = \tan x$


2. $latex \partial_y u + \partial_x u = u^2; u(x,0) = h(x)$

Problem 2

Find a smooth function $latex a(x,y)$ in $latex \R^2$ such that the equation

$latex \partial_y u + a(x,y) \partial_x u = 0$

does not have global solutions for any Cauchy data $latex \{y=0\}$.

Problem 3

Let $latex \alpha\in\R$ and $latex h(x)$ a continuous function in $latex \R$, and consider que problem

$latex y\partial_x u + x \partial_y u = \alpha u; \quad u(x,0) = h(x).$

1. Find all points where $latex \{y=0\}$ is characteristic.


2. What is the compatibility condition on h on those points?


3. Find the solution of the initial-value problem away of the point in (1).


4. What is the domain of this solution?


5. For the cases $latex \alpha=1, h(x) = x$ and $latex \alpha=3, h(x) = x$, check whether this solution can be extended over the characteristic points.


6. Find the characteristic curves containing the points in (1).


7. What is their relation to the domain in (4)?

Problem 4

Consider now the problem

$latex x \partial_x u + y \partial_y u = \alpha u; \quad u(x,0) = h(x).$

1. Check that $latex \{y=0\}$ is characteristic at every point.


2. Find all h satisfying the compatibility condition on $latex \{y=0\}$.


3. For $latex \alpha>0$, find two solutions with the given initial value on $latex \{y=0\}$.

Problem 5

In the plane, find two solutions of the initial-value problem

$latex x\partial_x u + y\partial_y u + \dfrac{1}{2}\big( (\partial_x u)^2 + (\partial_y u)^2\big) = u; \quad u(x,0) = \dfrac{1}{2}(1 - x^2).$