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

Meiirkhan B. Borikhanov; Michael Ruzhansky; Berikbol T. Torebek

doi:10.3934/dcdss.2024175

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2025, Volume 18, Issue 8: 2127-2141. Doi: 10.3934/dcdss.2024175

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

1.
Khoja Akhmet Yassawi International Kazakh-Turkish University, Sattarkhanov ave., 29, 161200 Turkistan, Kazakhstan
2.
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
3.
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Belgium
4.
School of Mathematical Sciences, Queen Mary University of London, United Kingdom

^*Corresponding author: Berikbol T. Torebek
^*Corresponding author: Berikbol T. Torebek

Received: April 2024

Revised: September 2024

Early access: September 25, 2024

Published online: September 25, 2024

This research has been funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP14869090), by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). Michael Ruzhansky is also supported by EPSRC grant EP/R003025/2 and EP/V005529/1.

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Abstract

In this paper, we study two nonlinear Sobolev-type equations on the Heisenberg groups. 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} $.

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Mathematics Subject Classification: Primary: 35R03, 35B33; Secondary: 35B44.

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