How do I solve this reaction-diffusion Cauchy problem?
$$\left\{ \begin{array}{l l} u_{t} - \kappa u_{xx} +ru=0 & \quad \mbox{$x \in \mathbb{R}, t>0$,}\\ \quad u(x,0) = \phi(x), \end{array} \right. $$
where $\kappa > 0$ and $r>0$ are constants.
So I defined $v(x,t)=e^{rt}u(x,t)$. How do I show that $v$ solves the heat equation with the same initial condition $\phi$? Then I think I find $u$ by solving the heat equation for $v$? But how do I go about this as well?
Thank you.
Note that the given equation
$u_t -\kappa u_{xx} + ru \tag 1$
may be re-arranged to read
$\kappa u_{xx} = u_t + ru; \tag 2$
if we set
$v = e^{rt}u; \tag 3$
then
$v_t = re^{rt}u + e^{rt}u_t, \tag 4$
and also
$v_{xx} = e^{rt}u_{xx}; \tag 5$
thus by (2) and (4),
$\kappa v_{xx} = \kappa e^{rt} u_{xx} = e^{rt}u_t + e^{rt}ru = v_t; \tag 6$
furthermore,
$v(x, 0) = e^{r(0)}u(x, 0) = u(x, 0) = \phi(x); \tag 7$
we see in (6) and (7) that $v(x,t)$ solves the heat equation with the same initial conditions as $u$.
As far as solving the equation
$v_t = \kappa v_{xx}, \; v(x, 0) = \phi(x) \tag 9$
is concerned, there are many techniques available; it is a big topic, covered in many texts on partial differential equations, as well as on many web sites. A good place to start is these wikipedia articles on the heat equation and the heat kernel. They probably give better information on how to solve (9) than I could here.