Find solutions of the following initial-value problems in .
Find a smooth function in such that the equation
does not have global solutions for any Cauchy data .
Let and a continuous function in , and consider que problem
- Find all points where is characteristic.
- What is the compatibility condition on h on those points?
- Find the solution of the initial-value problem away of the point in (1).
- What is the domain of this solution?
- For the cases and , check whether this solution can be extended over the characteristic points.
- Find the characteristic curves containing the points in (1).
- What is their relation to the domain in (4)?
Consider now the problem
- Check that is characteristic at every point.
- Find all h satisfying the compatibility condition on .
- For , find two solutions with the given initial value on .
In the plane, find two solutions of the initial-value problem