Bounds for Blow-up Time to a Viscoelastic Hyperbolic Equation of Kirchhoff Type with Variable Sources
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Bounds for Blow-up Time to a Viscoelastic Hyperbolic Equation of Kirchhoff Type with Variable Sources Menglan Liao1
· Bin Guo1 · Xiangyu Zhu1,2
Received: 22 December 2019 / Accepted: 30 August 2020 © Springer Nature B.V. 2020
Abstract The aim of this paper is to study bounds for blow-up time to the following viscoelastic hyperbolic equation of Kirchhoff type with initial-boundary value condition: |ut |ρ utt − M(∇u22 )u +
t
g(t − τ )u(τ )dτ + |ut |m(x)−2 ut = |u|p(x)−2 u. 0
Compared with constant exponents, it is difficult to discuss the above problem due to the existence of a gap between the modular and the norm. The authors construct suitable function spaces to discuss the upper bound for blow-up time with positive initial energy by means of a differential inequality technique. In addition, lower bounds for blow-up time in different range of exponent are obtained. These improve and generalize some recent results. Keywords Blow-up · Kirchhoff · Variable sources · Viscoelasticity · Positive initial energy · Damping term Mathematics Subject Classification (2010) 35L20 · 35L70 · 35B44 The project is supported by NSFC (11301211), by the Scientific and Technological Project of Jilin Province’s Education Department in Thirteenth-five-year (JJKH20180111KJ).
B B. Guo
[email protected] M. Liao [email protected] X. Zhu [email protected]
1
School of Mathematics, Jilin University, Changchun, Jilin Province 130012, China
2
Department of Mathematics, College of Science, Yanbian University, Yanji, Jilin Province 133002, China
M. Liao et al.
1 Introduction In this paper, we consider the following viscoelastic hyperbolic equation of Kirchhoff type: |ut |ρ utt − M(∇u22 )u + = |u|
p(x)−2
t
g(t − τ )u(τ )dτ + |ut |m(x)−2 ut 0
(1.1)
u (x, t) ∈ × (0, T )
with initial–boundary value u(x, t) = 0 (x, t) ∈ ∂ × (0, T ), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) x ∈ , where is a bounded smooth domain in RN (N ≥ 3), 0 < T < ∞, u0 (x) ∈ H01 (), u1 (x) ∈ L2 (). M(s) = m0 + bs γ is a positive C 1 function with parameters m0 > 0, b ≥ 0, s ≥ 0, γ ≥ 1. The exponents m(x) and p(x) are continuous functions on with the logarithmic module of continuity: ∀ x, y ∈ , |x − y| < 1, |m(x) − m(y)| + |p(x) − p(y)| ≤ ω(|x − y|),
(1.2)
where lim sup ω(τ ) ln
τ →0+
1 = C < ∞. τ
In addition to this condition, the exponents satisfy the following: 2 ≤ m− := ess inf m(x) ≤ m(x) ≤ m+ := ess sup m(x) ≤ x∈
x∈
2 < p − := ess inf p(x) ≤ p(x) ≤ p + := ess sup p(x) ≤ x∈
x∈
0 0, g (τ ) ≤ 0, ∞ m0 − g(τ )dτ = k > 0.
(1.6) (1.7)
0
Problem (1.1) comes from the original equation below: ∂u Eh ∂ 2u = p0 + ρh 2 + δ ∂t ∂t 2L
0
L
∂u 2 ∂ 2 u dx + f, ∂x ∂x 2
0 ≤ x ≤ L, t ≥ 0,
where u = u(x, t) is the lateral deflection, E is the Young’s modulus, ρ is the mass density, h is the cross-section area, L is the length, p0 is the initial axial tension, δ is the resistance modulus, and f is the external force. When δ = f = 0, Kirchhoff [1] first introduced the above problem to describe the nonlinear vibrations of a
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