UMD PDE Qualifying Exams/EvansCh2

Problem 1Edit

Write an explicit formula for a function   solving the initial-value problem

 


where   and   are constants.

SolutionEdit

Consider characteristics  . Also, for any  , consider  . Then taking a derivative gives

 

where the last inequality is a result of the original PDE. The above ODE can be solved and we get  

Finally, any point   is connected to the characteristic curve   where   and hence

 .