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We study the problems of uniqueness for Hardy-H\'enon parabolic equations,which are semilinear heat equations with the singular potential (Hardy type) orthe increasing potential (H\'enon type) in the nonlinear term. To deal with theHardy-H\'enon type nonlinearities, we employ weighted Lorentz spaces assolution spaces. We prove unconditional uniqueness and non-uniqueness, and weestablish uniqueness criterion for Hardy-H\'enon parabolic equations in theweighted Lorentz spaces. The results extend the previous works on the Fujitaequation and Hardy equations in Lebesgue spaces.

Unconditional uniqueness and non-uniqueness for Hardy-H\'enon parabolic equations