Instantaneous blow up solutions for nonlinear Sobolev type equations on the Heisenberg Group

TL;DR

This work proves instantaneous blow-up for nonlinear Sobolev type equations on the Heisenberg group $\mathbb{H}^{n}$. It employs the nonlinear capacity method with carefully designed test functions to show the Cauchy problems have no nontrivial local weak solutions when $1<q\leq q_c$, where $q_c=\frac{Q}{Q-2}$ and $Q=2n+2$. The critical exponent is shown to be optimal, since for $q>q_c$ stationary supersolutions exist, preventing blow-up for small data. The analysis combines Heisenberg group geometry, sub-Laplacian invariances, and energy-type estimates to establish nonexistence of local weak solutions for the Cauchy problems in this sub-Riemannian setting.

Abstract

In this paper, we study the nonlinear Sobolev type equations on the Heisenberg group. We show that the problems do not admit nontrivial local weak solutions, i.e. "instantaneous blow up" occurs, using the nonlinear capacity method. Namely, by choosing suitable test functions, we will prove an instantaneous blow up for any initial conditions $u_0,\,u_1\in L^q(\mathbb{H}^n)$ with $q\leq \frac{Q}{Q-2}$.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• proof : Proof of Lemma \ref{['LT1']}.
• Lemma 2.2
• proof : Proof of Lemma \ref{['LT3']}.
• Definition 3.1
• Definition 3.2
• proof : Proof of Theorem \ref{['T1']}.