Let be the soloution of that is defined in the whole plane. Prove that .

Hint: Think of the characteristic curves of this PDE.

HOPE You'll be able to help me

Thanks in advance!

Printable View

- November 2nd 2010, 02:29 AMWannaBePDE-Characteristic Curves
Let be the soloution of that is defined in the whole plane. Prove that .

Hint: Think of the characteristic curves of this PDE.

HOPE You'll be able to help me

Thanks in advance! - November 4th 2010, 03:20 AMRebesques
- November 4th 2010, 07:17 AMWannaBe
I've already done this...We'll get:

and: where is k is a function of y ...

We'll have a discontinuity every time that and every time that x=0...

How did you deduced the concloution that the discontinuities are at the origin?

Thanks ! - November 4th 2010, 09:27 AMRebesques
From y=c_1x. Thus there will always be a discontinuity at the origin, but the expression lnx+k(y) may be nonzero for a suitable choice of initial data.

- November 4th 2010, 10:42 AMWannaBe
is a line segmant in ...Why there is a discontinuity at the origin?

- November 4th 2010, 12:07 PMRebesques
It is one of the characteristics though. The constant c_1 is the slope and will vary along each characteristic, while being (explicitly) determined by c_1=y/x. So at x=0 (for which y=0) the slope will become infinite and the solution will develop a shock. check your notes.

- November 4th 2010, 12:57 PMWannaBe
actually, at x=0, the expression ln(x) has a discontinuity... So we can deduce the result from this (I think...)...

Thanks...I think I understand it