Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients

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March 2006, 5(1): 155-179. doi: 10.3934/cpaa.2006.5.155

Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients

Luisa Moschini ^1, , Guillermo Reyes ^2, and Alberto Tesei ^3,

1.	Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza” P.le A. Moro 5, I-00185, Roma, Italy
2.	Depto. de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avda. de la Universidad 30, Leganés 28911, Madrid, Spain
3.	Dipartimento di Matematica "G. Castelnuovo", Università di Roma "La Sapienza", P.le A. Moro 5, I-00185 Roma

Received March 2005 Revised September 2005 Published December 2005

Abstract
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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.

Keywords: singular coefficients, inverse-square potential, nonuniqueness of solutions, weighted Sobolev spaces., Semilinear parabolic equations.

Mathematics Subject Classification: Primary: 35K55; Secondary: 35K65, 34A1.

Citation: Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155

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