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

Vladimir Georgiev; Alessandro Palmieri

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

Graduate School of Advanced Science and Engineering

Research output: Contribution to journal › Article › peer-review

Abstract

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 p_Fuj(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 > p_Fuj(Q) in an exponential weighted energy space. On the other hand, a blow-up result for 1 < p ≤ p_Fuj(Q) under certain integral sign assumptions for the Cauchy data by using the test function method.

Original language	English
Journal	Unknown Journal
Publication status	Published - 2019 Aug 8

Keywords

Critical exponent
Damped wave equation
Energy spaces with exponential weight
Heisenberg group
Test function method

ASJC Scopus subject areas

General

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title = "Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity",

abstract = "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 1 < p ≤ pFuj(Q) under certain integral sign assumptions for the Cauchy data by using the test function method.",

keywords = "Critical exponent, Damped wave equation, Energy spaces with exponential weight, Heisenberg group, Test function method",

author = "Vladimir Georgiev and Alessandro Palmieri",

year = "2019",

month = aug,

day = "8",

language = "English",

journal = "Nuclear Physics A",

issn = "0375-9474",

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AU - Georgiev, Vladimir

AU - Palmieri, Alessandro

PY - 2019/8/8

Y1 - 2019/8/8

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 1 < p ≤ pFuj(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 1 < p ≤ pFuj(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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