The laplace equation whit initial conditions(adsbygoogle = window.adsbygoogle || []).push({});

u_tt + u_xx = 0 -oo<x<oo , t>0

u(x,0)=0

u_t(x,0)=f_k(x)

where f_k(x)=sin(kx), has the unique solution

u(x,t)=(1/k)sin(kx)sin(kt)

Show that the problem is ill posed.

I know that the equations is elliptic so i tried first whit the maximum principle but

this Partial differential equation has no boundary condition so i can use that principle.

The fourier method requires a periodic boundary condition, but again there is no boundary condition in this PDE.

I then tried the energy method and i get this:

d||u(*,t)||^2/dt=d/dt (int (u(x,t)^2)) after some work i get (2k/tan(kx))||u(*,t)||^2 but how does this show that the problem is ill posed? am i doing it right? if not the how do i do? thx :D

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# Help whit ill-posedness

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