Nonexistence Results for the Hyperbolic-Type Equations on Graded Lie Groups
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Nonexistence Results for the Hyperbolic-Type Equations on Graded Lie Groups Aidyn Kassymov1,2,3 · Niyaz Tokmagambetov1,2,4
· Berikbol Torebek1,2,4
Received: 23 September 2019 / Revised: 18 February 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we deal with systems of wave and pseudo-hyperbolic equations. Some semilinear equations for hypoelliptic operators on Rn , Heisenberg groups, stratified Lie groups and graded Lie groups are studied. In particular, we obtain nonexistence results for nonlinear hyperbolic and pseudo-hyperbolic equations and systems on graded Lie groups. Also, we show Kato-type exponents for systems of pseudo-hyperbolic equations for Rockland operators on graded Lie groups. Keywords Rockland operator · Graded Lie groups · Blow-up · Kato-type exponent · Pseudo-hyperbolic equation Mathematics Subject Classification 35R03
Communicated by Norhashidah Hj. Mohd. Ali. The authors were supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations, as well as by the MESRK Grants AP08053051 and AP08052001 of the Ministry of Education and Science of the Republic of Kazakhstan. NT and BT were supported in parts by the RUDN University Program “5-100”.
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Niyaz Tokmagambetov [email protected] Aidyn Kassymov [email protected]; [email protected] Berikbol Torebek [email protected]
1
Al-Farabi Kazakh National University, 71 Al-Farabi Avenue, 050040 Almaty, Kazakhstan
2
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000 Ghent, Belgium
3
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
4
Peoples’ Friendship University of Russia, RUDN University, Moscow, Russia
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A. Kassymov et al.
1 Introduction In 1979, John [13,14] started to study the nonlinear wave equation ∂ 2 u(x, t) − u(x, t) = |u(x, t)| p , (x, t) := R N × (0, +∞), ∂t 2
(1.1)
for N > 1 and p > 1, with the Cauchy data u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ R N . After 2 years, Strauss [27] suggested that the critical exponent of the existence of a global solution to the problem (1.1) is the largest root of the quadratic equation (N − 1) p 2 − (N + 1) p − 2 = 0.
(1.2)
Later, Sideris [28] proved that if u 0 , u 1 ∈ C0∞ (R N ) are sufficiently small and satisfy some weak positivity conditions, then the corresponding solution cannot be global if 1 < p < p ∗ (N ), where p ∗ is the larger root of (1.2). Note that in the case of N = 1, the problem (1.1) has no global solution. The nonexistence of a global√ solution of problem (1.1) was proved by Glassey [9] for N = 2 and 1 < p ≤ 2+2 17 , and by John [13] and by Schaeffer [26] for N = 3 and √ 1 < p ≤ 1 + 2. Using a significantly different method, Kato [15] generalized the above results with more general assumptions. For the wave problem (1.1), Kato’s result states that if u is a generalized solution of problem (1.1) with u 0 , u 1 ∈ C0∞ (R N ), supp u ⊂ {|x| ≤ R + t} and
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