L∞ blow-up in the fourth-order Jordan-Moore-Gibson-Thompson equation
Abstract
We investigate the fourth-order Jordan-Moore-Gibson-Thompson equation utttt+αuttt+βutt−γΔutt−δΔut−ϱ Δu=(f(u))ttt+(f(u))tt under homogeneous Dirichlet boundary conditions in a smooth bounded domain Ω⊂ ℝn with n≤ 3, where α∈ ℝ and β,γ,δ,ϱ>0. First, for any suitably regular initial data and assuming that f∈ C4(ℝ) with f(0)=0, we prove that the corresponding initial-boundary value problem admits a unique maximal strong solution u on (0, Tmax). Second, by combining three time-integrated formulations with a series of resolvent-regularized energy estimates, we show that gradient blow-up is ruled out in the sense that tag* Tmax<∞ ⇒ limsuptnearrow Tmax|u(·,t)|L∞(Ω)=∞. Finally, assuming in addition that f is convex and grows superlinearly in the sense that f''(ξ)≥ 0 for all ξ∈ℝ, f(ξ)/ξ→ +∞ as ξ→ +∞ , ∫ξ0∞dξ/f(ξ)<∞ for some ξ0>0, we employ the extensibility criterion (*) to prove that, for a suitable class of initial data, the solution blows up in finite time in the L∞-norm. To the best of our knowledge, this provides the first rigorous description of finite-time blow-up for a genuinely nonlinear equation of fourth order in time.
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BibTeX
@misc{boudjeriou2026blowup,
title = {{$L^\infty$ blow-up in the fourth-order Jordan-Moore-Gibson-Thompson equation}},
author = {Tahir Boudjeriou and Filippo Dell'Oro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33213},
eprint = {2610.33213},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33213},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - L^∞ blow-up in the fourth-order Jordan-Moore-Gibson-Thompson equation
AU - Tahir Boudjeriou
AU - Filippo Dell'Oro
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.33213
UR - https://arxiv.org/abs/2610.33213
AB - We investigate the fourth-order Jordan-Moore-Gibson-Thompson equation u_tttt+αu_ttt+βu_tt−γΔu_tt−δΔu_t−ϱ Δu=(f(u))_ttt+(f(u))_tt under homogeneous Dirichlet boundary conditions in a smooth bounded domain Ω⊂ ℝ^n with n≤ 3, where α∈ ℝ and β,γ,δ,ϱ>0. First, for any suitably regular initial data and assuming that f∈ C^4(ℝ) with f(0)=0, we prove that the corresponding initial-boundary value problem admits a unique maximal strong solution u on (0, T_max). Second, by combining three time-integrated formulations with a series of resolvent-regularized energy estimates, we show that gradient blow-up is ruled out in the sense that tag* T_max<∞ ⇒ limsup_tnearrow T_max|u(·,t)|_L^∞(Ω)=∞. Finally, assuming in addition that f is convex and grows superlinearly in the sense that f''(ξ)≥ 0 for all ξ∈ℝ, f(ξ)/ξ→ +∞ as ξ→ +∞ , ∫_ξ_0^∞dξ/f(ξ)<∞ for some ξ_0>0, we employ the extensibility criterion (*) to prove that, for a suitable class of initial data, the solution blows up in finite time in the L^∞-norm. To the best of our knowledge, this provides the first rigorous description of finite-time blow-up for a genuinely nonlinear equation of fourth order in time.
ER -