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Non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data

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Abstract.

The Cauchy problem for semilinear heat equations with singular initial data is studied, where N≥2, λ>0 is a parameter, and a≥0, a≠0. We show that when p>(N+2)/N and (N−2)p<N+2, there exists a positive constant such that the problem has two positive self-similar solutions and with if and no positive self-similar solutions if . Furthermore, for each fixed and in L ∞(R N) as λ→0, where w 0 is a non-unique solution to the problem with zero initial data, which is constructed by Haraux and Weissler.

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  1. Department of Applied Mathematics, Faculty of Engineering, Kobe University, Nada, Kobe 657-8501, Japan

    Yūki Naito

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  1. Yūki Naito
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Correspondence to Yūki Naito.

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Mathematics Subject Classification (2000): 35K55, 35J60

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Naito, Y. Non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data. Math. Ann. 329, 161–196 (2004). https://doi.org/10.1007/s00208-004-0515-4

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