I wanted to solve the following PDE with initial condition

$$ \left\{\begin{array}{c} xu_t+u_x=0\\ u(0,x)=f(x) \end{array}\right.$$

Proving that:

(i) if $f(x) = \sin(x),$ then it is impossible to find a solution which is valid for all point in $\mathbb{R}^2.$

(ii) if $f(x) = \cos(x),$ has infinite solutions defined in all $\mathbb{R}^2.$

I tried to solve this using the method of characteristics.

First of all the characteristic system is

$$ \left\{\begin{array}{c} \frac{d t}{d s} = x\\ \frac{d x}{d s}=1 \\ \frac{d u}{d s}=0 \end{array}\right.$$ with initial conditions $$ t(0,\tau)=0 $$ $$ x(0,\tau)=\tau $$ $$ u(0,\tau)=\sin \tau $$

Computing, and proceeding by an analogue method to $\cos(x),$ I find $u(t,x) = \sin (\sqrt{-2t + x^2})$ and $u(t,x) = \cos (\sqrt{-2t + x^2}),$ but both are not defined in $(1,0).$ There is a problem with my calculation, or the initial statement are wrong?

  • $\begingroup$ The radicand not defined for some values of the variables is important for the second question. $\endgroup$ – Rafa Budría Mar 16 at 21:06

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