TY - JOUR

T1 - Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity

AU - Georgiev, Vladimir

AU - Palmieri, Alessandro

N1 - Funding Information:
V. Georgiev is supported in part by GNAMPA - Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni , by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences , by Top Global University Project, Waseda University and by the University of Pisa , Project PRA 2018 49 . A. Palmieri is supported by the University of Pisa , Project PRA 2018 49 .

PY - 2020/6/15

Y1 - 2020/6/15

N2 - In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power non-linearity. We prove that the critical exponent is the Fujita exponent pFuj(Q)=1+2/Q, where Q is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for p>pFuj(Q) in an exponential weighted energy space. On the other hand, a blow-up result for 1Fuj(Q) under certain integral sign assumptions for the Cauchy data by using the test function method.

AB - In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power non-linearity. We prove that the critical exponent is the Fujita exponent pFuj(Q)=1+2/Q, where Q is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for p>pFuj(Q) in an exponential weighted energy space. On the other hand, a blow-up result for 1Fuj(Q) under certain integral sign assumptions for the Cauchy data by using the test function method.

KW - Critical exponent

KW - Damped wave equation

KW - Energy spaces with exponential weight

KW - Heisenberg group

KW - Test function method

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U2 - 10.1016/j.jde.2019.12.009

DO - 10.1016/j.jde.2019.12.009

M3 - Article

AN - SCOPUS:85077167720

VL - 269

SP - 420

EP - 448

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -