Blow-up for coupled systems of semilinear damped wave equations on the Heisenberg group

Dongmei Li; Sen Ming; Tianyu Liu; Dongmei Li; Sen Ming; Tianyu Liu

doi:10.3934/era.2026115

Electronic Research Archive

2026, Volume 34, Issue 4: 2483-2510. doi: 10.3934/era.2026115

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Research article

Blow-up for coupled systems of semilinear damped wave equations on the Heisenberg group

Department of Mathematics, North University of China, Taiyuan 030051, China

Received: 26 November 2025 Revised: 04 February 2026 Accepted: 04 March 2026 Published: 23 March 2026

The main purpose of this paper is to study behaviors of solutions to the Cauchy problem for coupled systems of semilinear damped wave equations with power nonlinearities on the Heisenberg group, where the Fujita critical exponent is determined. By means of the Bihari inequality as well as the Gagliardo-Nirenberg-type inequality and the contraction mapping principle, the local well-posedness of the Cauchy problem is successfully proved. Combining energy estimates and decay estimates together with a contradiction argument, the global existence of solutions is also demonstrated. At the same time, the blow-up results and upper bound estimation of the solutions is derived by using the test function technique. Our main new contribution is the derivation of a fundamental inequality utilized in weighted energy estimates, as well as the selection of exponents in the scaling of two convex functions for constructing test functions.
- coupled damped wave equations,
- Heisenberg group,
- well-posedness,
- test function method,
- blow-up
Citation: Dongmei Li, Sen Ming, Tianyu Liu. Blow-up for coupled systems of semilinear damped wave equations on the Heisenberg group[J]. Electronic Research Archive, 2026, 34(4): 2483-2510. doi: 10.3934/era.2026115

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Abstract

The main purpose of this paper is to study behaviors of solutions to the Cauchy problem for coupled systems of semilinear damped wave equations with power nonlinearities on the Heisenberg group, where the Fujita critical exponent is determined. By means of the Bihari inequality as well as the Gagliardo-Nirenberg-type inequality and the contraction mapping principle, the local well-posedness of the Cauchy problem is successfully proved. Combining energy estimates and decay estimates together with a contradiction argument, the global existence of solutions is also demonstrated. At the same time, the blow-up results and upper bound estimation of the solutions is derived by using the test function technique. Our main new contribution is the derivation of a fundamental inequality utilized in weighted energy estimates, as well as the selection of exponents in the scaling of two convex functions for constructing test functions.

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