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

• Published in: Journal of Differential Equations Vol. 269; no. 1; pp. 420 - 448
• Main Authors: Georgiev, Vladimir, Palmieri, Alessandro
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.06.2020
• Subjects: Critical exponent; Damped wave equation; Energy spaces with exponential weight; Heisenberg group; Test function method; secondary; Heisenberg group; Energy spaces with exponential weight; Damped wave equation; Critical exponent; primary; Test function method
• ISSN: 0022-0396; 1090-2732; 1090-2732
• DOI: 10.1016/j.jde.2019.12.009

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.