2024/04/02 by Khaldi Said, Said, Khaldi, Arioui Fatima Zahra +3
Computer Science · Engineering · Mathematics · #35B45 #35G10 #42B10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2404.01544
openalex publication_date 2024/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Our aim in this paper is to discuss the critical exponent in semi-linear structurally damped wave and beam equations with additional dispersion term. The special model we have in mind is utt(t,x)+(-Δ)σu(t,x)+(-Δ)2δu(t,x)+2(-Δ)δut(t,x)=|u(t,x)| p where the initial displacement u(0,x)=u0(x), the initial velocity ut(0,x)=u1(x) and the parameters t∈ [0,∞), x∈ ℝn, σ≥ 1, δ∈(0,\fracσ2), p>1. The solution to the linear equation at low frequency region involves an interplay of diffusion and oscillation phenomena represented by a real-complex Fourier multiplier of the form m(t,ξ)=\frace^-|ξ|2δt± i|ξ|σt2i|ξ|σ, ξ∈ ℝn, i=√(-1). The scaling argument shows that the diffusive part leads to faster decay rates compared to the oscillatory one. This interplay creates a new gap in the critical exponent between the blow up (in finite time) result when 11+(σ+2δ)/(n-σ) (super-critical case). We leave an open to show if this gap will be closed at least in low or high space dimensions because, to the best of authors knowledge, the necessary Fourier multiplier that leads to the sub-critical case does not explicitly appear in m(t,ξ).