Apologies for my overly simple problem.
I am looking at the generic diffusion-decay PDE
$u_t=D\nabla^2u-\delta u(x,y,t),~u(0,0,t)=u_0,$
and I am interested in the steady-state profile of $u(x,y,t)$, i.e. a solution to
$0=D\nabla^2u-\delta u(x,y,t),~u(0,0)=u_0.$
Using the ansatz
$u(x,y,t) = F e^{\lambda x} + G e^{-\lambda x} + J e^{\lambda y} + K e^{-\lambda y},$
I find one possible solution with
$\lambda^2=\delta/D,~F=G=J=K=u_0/4.$
The problem I have now is that this solution grows away from the origin which I find puzzling as I expected to find a solution that has a peak in the origin and decays away from it.
Could anyone point me in the right direction please?
Thank you.
Edit:
Apologies for my slow response but @Andrew's answer was a little intimidating (and still is, although I don't seem to see it now - did they delete their answer?) and so I had to do a little background reading.
Thanks @Willie Wong for your answer and for giving me some intuition as to why my expectation is wrong. Also thanks to @Andrew for pointing me in the direction of fundamental solutions and so forth.
Following @Andrew I found these lecture notes: http://www.stanford.edu/class/math220b/handouts/laplace.pdf
Where they construct a radial solution for the Laplace equation using the ansatz
$u(\mathbf{x},t)=v(|\mathbf{x}|,t),$
and knowing the derivative of the absolute value function and defining radius $r=|\mathbf{x}|$ I get:
$v_t=D(v_{rr} + \frac{1}{r} v_r) - \delta v~;~D,\delta>0~;~r>0$
For the steady state equation
$0=D(v_{rr} + \frac{1}{r} v_r) - \delta v~;~D,\delta>0~;~r>0$
I use the ansatz
$v(r)=v_0 \exp{\lambda r},$
which gives me
$0=D v_0 \exp{(\lambda r)} (\lambda^2 + \frac{\lambda}{r} - \delta)$
so using $D>0$, $v_0> 0$ I get
$\lambda_{1,2}=-\frac{1}{2r} \mp \frac{1}{2} \sqrt{r^{-2}+4 \delta / D}.$
Of course this solution blows up towards the origin so I am again a little puzzled and would appreciate any advice / help!
In those notes I linked above, they construct the fundamental solution (starting around Eqn 3.2) which is probably what I want but I don't yet understand how some of the steps work.