$\partial_t u+\partial_x u+u=0$

Question

I am trying to solve $\partial_t u+\partial_x u+u=0$. It looks like transport equation $\partial_t u+b\cdot Du=0$. But the idea for solving the transport equation is by noticing a particular directional derivative of $u$ vanishes, but this kind of idea does not apply here. Any help will be appreciated.

Answer 1

The same basic idea - the method of characteristics - may work here. Fix $x_0$, let $v(s) = u(s, x_0+s)$. We have $$ v'(s) = \partial_t u(s, x_0+s) + \partial_x u(s, x_0+s) = -u(s, x_0+s) = -v(s) $$ giving you an ode for $v$. The solution to $v'=-v$ is $v(s) = \exp(-s)v(0)$. Hence, we have $$ u(s, x_0 + s) = \exp(-s)u(0, x_0) $$ Now, for a given $(t, x)$, let $x_0 = x-t$. We then have $$ u(t, x) = u(t, x_0+t) = \exp(-t)u(0, x-t). $$