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Existence and non-existence of global solutions for a semilinear heat equation

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Abstract

The existence and non-existence of global solutions and theL p blow-up of non-global solutions to the initial value problemu′(t)=Δu(t)+u(t)γ onR n are studied. We consider onlyγ>1. In the casen(γ − 1)/2=1, we present a simple proof that there are no non-trivial global non-negative solutions. Ifn(γ−1)/2≦1, we show under mild technical restrictions that non-negativeL p solutions always blow-up inL p norm in finite time. In the casen(γ−1)/2>1, we give new sufficient conditions on the initial data which guarantee the existence of global solutions.

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Authors and Affiliations

  1. Department of Mathematics, Brown University, 02912, Providence, RI, USA

    Fred B. Weissler

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  1. Fred B. Weissler
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Research partially supported by NSF grant MCS79-03636.

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Weissler, F.B. Existence and non-existence of global solutions for a semilinear heat equation. Israel J. Math. 38, 29–40 (1981). https://doi.org/10.1007/BF02761845

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  • DOI: https://doi.org/10.1007/BF02761845

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