Solutions’ Existence for A Hyperbolic Nonlinear Wave Equation of Kirchhoff Type in Unbounded Domains and Estimation of The Upper Bound for The Blow-Up Time of The Solutions
DOI:
https://doi.org/10.65904/3083-1733.2026.2.8Keywords:
Hyperbolic wave equation, Existence of solutions, Blow-up results, Quasilinear wave euation, Dissipative Kirchhoff’s stringAbstract
Our aim in this work is to study the existence of global solutions and blow-up phenomena in finite time of the following quasilinear dissipative Kirchhoff’s string problem:
utt − φ(x)‖∇u(t)‖² Δu + δu_t = f, x ∈ ℝᴺ, t ≥ 0
with initial conditions u(x,0) = u₀(x) and ut(x,0) = u₁(x), where (φ(x))⁻¹ ≡ g(x) ∈ L(N/2)(ℝᴺ) ∩ L^∞(ℝᴺ), N ≥ 3, δ ≥ 0 the resistance modules and f ≡ |u|^a u the external force.
A combination of the Faedo-Galerkin approximation and the Banach Fixed-Point Theorem is used to estimate the local (unique) existence of the solutions, while the method of the modified potential well is used to prove the global existence and energy estimates of the solutions.
We complete our work with the blow-up analysis of the solutions for initial data of negative energy using the concavity method, where for the discrete case a = 2 we prove the modification of the upper bound of time.
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