On the Heisenberg group, the semilinear damped wave equation with |u|^p has Fujita critical exponent p_Fuj(Q)=1+2/Q: global small-data solutions for p above it, blow-up for p at or below it.
Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisenberg group
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
In this paper we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group $\mathbf{H}_n$. The heat operator is given in this case by $\partial_t-\Delta_H$, where $\Delta_H$ is the so-called sub-Laplacian on $\mathbf{H}_n$. We prove that the Fujita exponent $1 + 2/Q$ is critical, where $Q = 2n + 2$ is the homogeneous dimension of $\mathbf{H}_n$. 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 $L^\infty$ space.
fields
math.AP 1years
2019 1verdicts
CONDITIONAL 1representative citing papers
citing papers explorer
-
Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity
On the Heisenberg group, the semilinear damped wave equation with |u|^p has Fujita critical exponent p_Fuj(Q)=1+2/Q: global small-data solutions for p above it, blow-up for p at or below it.