Nonuniqueness of solutions  to  semilinear parabolic  equations with singular coefficients

Luisa Moschini; Guillermo Reyes; Alberto Tesei

doi:10.3934/cpaa.2006.5.155

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2006, Volume 5, Issue 1: 155-179. Doi: 10.3934/cpaa.2006.5.155

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Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients

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

Received: February 28, 2005

Revised: August 31, 2005

Published: February 2006

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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.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35K65, 34A12.

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