On the generalized parabolic Hardy-Hénon equation: Existence, blow-up, self-similarity and large-time asymptotic behavior

Abstract

This paper deals with the Cauchy problem for the Hardy-Hénon 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 Cb([0,∞); L c(R)). As a direct consequence, global existence for data in strong critical Lebesgue Lc (R) 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 ≡ 0 is shown to be asymptotically stable in Lc (R) – it is the only self-similar solution which is initially small in Lc (R). Moreover, blow-up results are obtained under mild assumptions on the initial data and the corresponding Fujita critical exponent is found.