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 .
Publisher Copyright:
© 2019 Elsevier Inc.
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
SN - 0022-0396
VL - 269
SP - 420
EP - 448
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 1
ER -