Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients

Abstract

We prove nonuniqueness of solutions of the Cauchy problem for a semilinear parabolic equation with inverse-square potential in certain Lebesgue spaces. The nonuniqueness results proved in [5] are the limiting case of the present ones as the strength of the potential vanishes. Similar results are obtained for a related semilinear parabolic equation with singular coefficients. The proofs rely on investigating by variational methods in suitable weighted Sobolev spaces the equation satisfied by the profile of a radial similarity solution.