Existence of solutions semilinear parabolic equations with singular initial data in the Heisenberg group

TL;DR

This paper establishes conditions for the existence of solutions to fractional semilinear heat equations with singular initial data in the Heisenberg group, identifying critical exponents and optimal singularity strength for solvability.

Contribution

It provides necessary and sufficient conditions for solvability, determines the Fujita-exponent in the Heisenberg group, and characterizes the initial data's singularity impact.

Findings

Identified the Fujita-exponent as 1+2/Q in the Heisenberg group.

Established optimal singularity strength for local-in-time solvability.

Derived sharp lifespan estimates for solutions with polynomial decay.

Abstract

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of fractional semilinear heat equations with power nonlinearities in the Heisenberg group $\mathbb{H}^N$. Using these conditions, we can prove that $1+2/Q$ separates the ranges of exponents of nonlinearities for the global-in-time solvability of the Cauchy problem (so-called the Fujita-exponent), where $Q=2N+2$ is the homogeneous dimension of $\mathbb{H}^N$, and identify the optimal strength of the singularity of the initial data for the local-in-time solvability. Furthermore, our conditions lead sharp estimates of the life span of solutions with nonnegative initial data having a polynomial decay at the space infinity.