arXiv subjects

Mohammad A. Rammaha

Publications and source records attributed to Mohammad A. Rammaha.

9 recordsLinked to original sources

arXiv2026

On a structural acoustic model with logarithmic supercritical source terms

In this paper, we study a structural acoustic model consisting of a semilinear wave equation defined on a three-dimensional bounded domain, coupled with a Kirchhoff-Love plate equation acting on a flat portion of the boundary. Primarily for mathematical interest, we impose nonlinear damping terms and logarithmic-type supercritical source terms on the system. We investigate local and global well-posedness, energy decay rates of potential well solutions, and blow-up of solutions under different conditions on the parameters and initial data. The main novelties include the analysis of the interaction between nonlinear damping and logarithmic energy-amplifying source terms, as well as the development of techniques for handling logarithmic source terms within potential well theory. The logarithmic nonlinearities are not homogeneous under scaling, which creates difficulties in studying potential well solutions. The wave-plate coupling through the acoustic pressure also causes technical difficulties in the analysis, especially in the proof of blow-up of weak solutions.

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arXiv2023

Blow-up of a structural acoustics model

This article studies the finite time blow-up of weak solutions to a structural acoustics model consisting of a semilinear wave equation defined on a bounded domain $Ω\subset\mathbb{R}^3$ which is strongly coupled with a Berger plate equation acting on the elastic wall, namely, a flat portion of the boundary. The system is influenced by several competing forces, including boundary and interior source and damping terms. We stress that the power-type source term acting on the wave equation is allowed to have a supercritical exponent, in the sense that its associated Nemytskii operators is not locally Lipschitz from $H^1$ into $L^2$. In this paper, we prove the blow-up results for weak solutions when the source terms are stronger than damping terms, by considering two scenarios of the initial data: (i) the initial total energy is negative; (ii) the initial total energy is positive but small, while the initial quadratic energy is sufficiently large. The most significant challenge in this work arises from the coupling of the wave and plate equations on the elastic wall.

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arXiv2022

On the asymptotic behavior of solutions to a structure acoustics model

This article concerns the long term behavior of solutions to a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain $Ω\subset\mathbb{R}^3$ which is coupled with a Berger plate equation acting on a flat portion of the boundary of $Ω$. The system is influenced by several competing forces, in particular a source term acting on the wave equation which is allowed to have a supercritical exponent. Our results build upon those obtained by Becklin and Rammaha [8]. With some restrictions on the parameters in the system and with careful analysis involving the Nehari manifold we obtain global existence of potential well solutions and establish either exponential or algebraic decay rates of energy, dependent upon the behavior of the damping terms. The main novelty in this work lies in our stabilization estimate, which notably does not generate lower-order terms. Consequently, the proof of the main result is shorter and more concise.

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arXiv2019

Global solutions to a structure acoustic interaction model with nonlinear sources

This article focuses on a structural acoustic interaction system consisting of a semilinear wave equation defined on a smooth bounded domain $Ω\subset\R^3$ which is strongly coupled with a Berger plate equation acting only on a flat part of the boundary of $Ω$. In particular, the source terms acting on the wave and plate equations are allowed to have arbitrary growth order. We employ a standard Galerkin approximation scheme to establish a rigorous proof of the existence of local weak solutions. In addition, under some conditions on the parameters in the system, we prove such solutions exist globally in time and depend continuously on the initial data.

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arXiv2018

On wave equations of the $p$-Laplacian type with supercritical nonlinearities

This article focuses on a quasilinear wave equation of $p$-Laplacian type: \[ u_{tt} - Δ_p u -Δu_t = f(u) \] in a bounded domain $Ω\subset \mathbb{R}^3$ with a sufficiently smooth boundary $Γ=\partial Ω$ subject to a generalized Robin boundary condition featuring boundary damping and a nonlinear source term. The operator $Δ_p$, $2<p<3$, denotes the classical $p$-Laplacian. The interior and boundary terms $f(u)$, $h(u)$ are sources that are allowed to have a supercritical exponent, in the sense that their associated Nemytskii operators are not locally Lipschitz from $W^{1,p}(Ω)$ into $L^2(Ω)$ or $L^2(Γ)$. Under suitable assumptions on the parameters we provide a rigorous proof of existence of a local weak solution which can be extended globally in time, provided the damping terms dominates the corresponding sources in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.

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arXiv2018

Energy decay of a viscoelastic wave equation with supercritical nonlinearities

This paper presents a study of the asymptotic behavior of the solutions for the history value problem of a viscoelastic wave equation which features a fading memory term as well as a supercritical source term and a frictional damping term: \begin{align*} \begin{cases} u_{tt}- k(0) Δu - \int_0^{\infty} k'(s) Δu(t-s) ds +|u_t|^{m-1}u_t =|u|^{p-1}u, \quad \text{ in } Ω\times (0,T), \\ u(x,t)=u_0(x,t), \quad \text{ in } Ω\times (-\infty,0], \end{cases} \end{align*} where $Ω$ is a bounded domain in $\mathbb R^3$ with a Dirichlét boundary condition and $u_0$ represents the history value. A suitable notion of a potential well is introduced for the system, and global existence of solutions is justified provided that the history value $u_0$ is taken from a subset of the potential well. Also, uniform energy decay rate is obtained which depends on the relaxation kernel $-k'(s)$ as well as the growth rate of the damping term. This manuscript complements our previous work [Guo et al. in J Differ Equ 257, 3778-3812(2014), J Differ Equ 262, 1956-1979(2017)] where Hadamard well-posedness and the singularity formulation have been studied for the system. It is worth stressing the special features of the model, namely the source term here has a supercritical growth rate and the memory term accounts to the full past history that goes back to $-\infty$.

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arXiv2018

Local and global existence of solutions to a strongly damped wave equation of the $p$-Laplacian type

This article focuses on a quasilinear wave equation of $p$-Laplacian type: $$ u_{tt} - Δ_p u - Δu_t=0$$ in a bounded domain $Ω\subset\mathbb{R}^3$ with a sufficiently smooth boundary $Γ=\partialΩ$ subject to a generalized Robin boundary condition featuring boundary damping and a nonlinear source term. The operator $Δ_p$, $2 < p < 3$, denotes the classical $p$-Laplacian. The nonlinear boundary term $f (u)$ is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from $W^{1,p}(Ω)$ into $L^2(Γ)$. Under suitable assumptions on the parameters we provide a rigorous proof of existence of a local weak solution which can be extended globally in time provided the source term satisfies an appropriate growth condition.

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arXiv2016

Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities

We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical nonlinear source: \begin{align*} \begin{cases} u_{tt}- k(0) Δu - \int_0^{\infty} k'(s) Δu(t-s) ds + |u_t|^{m-1}u_t=|u|^{p-1}u, \;\;\;\;\; Ω\times (0,T), \\ u(x,t)=u_0(x,t), \quad \text{ in } Ω\times (-\infty,0], \end{cases} \end{align*} where $Ω$ is a bounded domain in $\mathbb R^3$ with a Dirichlét boundary condition. The relaxation kernel $k$ is monotone decreasing and $k(\infty)=1$. We study blow-up of solutions when the source is stronger than dissipations, i.e., $p> \max\{m,\sqrt{k(0)}\}$, under two different scenarios: first, the total energy is negative, and the second, the total energy is positive with sufficiently large quadratic energy. This manuscript is a follow-up work of the paper [30] in which Hadamard well-posedness of this equation has been established in the finite energy space. The model under consideration features a supercritical source and a linear memory that accounts for the full past history as time goes to $-\infty$, which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite-time memory.

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arXiv2014

Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.

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