  This paper deals with the Cauchy problem for the Hardy-H\'{e}non equation
(and its fractional analogue). Local well-posedness for initial data in the
class of continuous functions with slow decay at infinity is investigated.
Small data (in critical weak-Lebesgue space) global well-posedness is obtained
in $C_{b}([0,\infty);L^{q_c,\infty}(\RR^{n}))$. As a direct consequence, global
existence for data in strong critical Lebesgue $L^{q_c}(\RR^{n})$ follows under
a smallness condition while uniqueness is unconditional. Besides, we prove the
existence of self-similar solutions and examine the long time behavior of
globally defined solutions. The zero solution $u\equiv 0$ is shown to be
asymptotically stable in $L^{q_c}(\RR^{n})$ -- it is the only self-similar
solution which is initially small in $L^{q_c}(\RR^{n})$. Moreover, blow-up
results are obtained under mild assumptions on the initial data and the
corresponding Fujita critical exponent is found.
