TY - JOUR

T1 - Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisenberg group

AU - Georgiev, Vladimir

AU - Palmieri, Alessandro

N1 - Publisher Copyright:
© 2020, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/6

Y1 - 2021/6

N2 - In this paper, we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group Hn. The heat operator is given in this case by ∂t-ΔH, where ΔH is the so-called sub-Laplacian on Hn. We prove that the Fujita exponent 1 + 2 / Q is critical, where Q= 2 n+ 2 is the homogeneous dimension of Hn. Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical case), we employ a revisited test function method developed recently by Ikeda–Sobajima. On the other hand, to find the lower bound estimate for the lifespan, we prove a local in time result in weighted L∞ space.

AB - In this paper, we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group Hn. The heat operator is given in this case by ∂t-ΔH, where ΔH is the so-called sub-Laplacian on Hn. We prove that the Fujita exponent 1 + 2 / Q is critical, where Q= 2 n+ 2 is the homogeneous dimension of Hn. Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical case), we employ a revisited test function method developed recently by Ikeda–Sobajima. On the other hand, to find the lower bound estimate for the lifespan, we prove a local in time result in weighted L∞ space.

KW - Critical exponent of Fujita type

KW - Heisenberg group

KW - Lifespan estimates

KW - Semilinear heat equation

KW - Test function method

KW - Weighted L spaces

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U2 - 10.1007/s10231-020-01023-z

DO - 10.1007/s10231-020-01023-z

M3 - Article

AN - SCOPUS:85089144490

SN - 0373-3114

VL - 200

SP - 999

EP - 1032

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

IS - 3

ER -