Local uniform stability for the semilinear wave equation in inhomogeneous media with locally distributed Kelvin–Voigt damping

We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary subject to a local viscoelastic damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase‐space. As far as we know, this is the first stabilization result for a semilinear wave equation with localized Kelvin–Voigt damping.     

Small data global well‐posedness for the nonlinear wave equation with nonlocal nonlinearity

In this paper, we investigate the Cauchy problem of the nonlinear wave equation , where V(u) = μ|·|^?γ ? |u|², , 0 < γ < min(4, n) and n ≥ 3. We prove small data global well‐posedness for the radial data and for the general data with angular regularity. We also give an improved result of the Hartree equation with negative critical regularity. Copyright © 2012 John Wiley & Sons, Ltd.     

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: “When is the Laplace–Beltrami operator , , invertible?” We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103–120). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let be an open and closed subset of the boundary of M. We say that has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance from a point to is bounded uniformly in x (and hence, in particular, intersects all connected components of M). For manifolds with finite width, we prove a Poincaré inequality for functions vanishing on , thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincaré inequality then leads, as in the classical case to results on the spectrum of Δ with domain given by mixed boundary conditions, in particular, Δ is invertible for manifolds with finite width. The bounded geometry assumption then allows us to prove the well‐posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces , .     

Blow‐up of solution of wave equation with internal and boundary source term and non‐porous viscoelastic acoustic boundary conditions

In this paper we study the blow‐up of solution of a mixed problem associated to a nonlinear wave equation with dissipative and source term in a bounded domain of . On a boundary portion of the domain we consider a non‐porous viscoelastic acoustic boundary conditions to a non‐locally reacting boundary.     

Blow‐up phenomena for a class of fourth‐order nonlinear wave equations with a viscous damping term

This paper deals with the blow‐up phenomena for a class of fourth‐order nonlinear wave equations with a viscous damping term with Ω = (0,1) and α > 0. Here, f(s) is a given nonlinear smooth function. For 0 < α < p – 1, we prove that the blow‐up occurs in finite time for arbitrary positive initial energy and suitable initial data. This result extends the recent results obtained by Xu et al. (Applicable Analysis)(2013) and Chen and Lu (J. Math. Anal. Appl.)(2009). Copyright © 2015 John Wiley & Sons, Ltd.     

Energy decay rates for solutions of the wave equations with nonlinear damping in exterior domain

In this paper we study the behaviors of the energy of solutions of the wave equations with localized nonlinear damping in exterior domains.     

Finite time blow‐up for a kind of initial‐boundary value problem of semilinear damped wave equation

In this paper, we consider the initial‐boundary value problem of semilinear damped wave equation u _{t t} ?Δu + u _t =|u |^p with power in an exterior domain. Blow‐up result in a finite time will be established in higher dimensions (n ≥3), no matter how small the initial data are. A special test function will be constructed, and then, we obtain the blow‐up result by a contradiction argument. Copyright © 2016 John Wiley & Sons, Ltd.     

Global solution and asymptotic behavior for the variable coefficient beam equation with nonlinear damping

This paper is concerned with the initial‐boundary value problem for a variable coefficient beam equation with nonlinear damping. Such a model arises from the vertical deflections of a damped extensible elastic inhomogeneous beam whose density depends on time and position. By using the Faedo–Galerkin method and energy method, we obtain the existence and uniqueness of global strong solution. Furthermore, the exponential decay estimate for the total energy is also derived. Copyright © 2015 John Wiley & Sons, Ltd.     

On a wave equation with supercritical interior and boundary sources and damping terms

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well‐posed in the finite‐energy space H¹(Ω) × L₂(?Ω) with boundary data in L₂ due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non‐dissipative and are not locally Lipschitz operators from H¹(Ω) into L₂(Ω), or L₂(?Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.     

Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities

In this article, we study the blow‐up of the damped wave equation in the scale‐invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}?\mathrm{\Delta }u+\frac{\mu }{1+t}{u}_t=|u_t{|}^p+|u{|}^q,\text{in}{?}^N\times [0,\infty ),$$ with small initial data. For $\mu <\frac{N(q?1)}{2}$ and μ ∈ (0, μ_?) , where μ_? > 0 is depending on the nonlinearties' powers and the space dimension (μ_? satisfies $(q?1)\left((N+2{\mu }_{?}?1)p?2\right)=4$), we prove that the wave equation, in this case, behaves like the one without dissipation (μ = 0 ). Our result completes the previous studies in the case where the dissipation is given by $\frac{\mu }{{(1+t)}^{\beta }}{u}_t;\beta >1$, where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term $\frac{\mu }{1+t}{u}_t$ is important.     

Well‐posedness and energy decay of solutions to a wave equation with a general boundary control of diffusive type

In this paper, we study well‐posedness and asymptotic stability of a wave equation with a general boundary control condition of diffusive type. We prove that the system lacks exponential stability. Furthermore, we show an explicit and general decay rate result, using the semigroup theory of linear operators and an estimate on the resolvent of the generator associated with the semigroup.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

In this paper we will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove blow‐up results or global existence (in time) of small data energy solutions.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation,II

This paper is a continuation of our recent paper 8 . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L² scale with additional L¹ regularity for the data. In order to deal with this L² regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.     

Mathematical models for the non‐isothermal Johnson–Segalman viscoelasticity in porous media: stability and wave propagation

We present a nonlinear model for Johnson–Segalman type polymeric fluids in porous media, accounting for thermal effects of Oldroyd‐B type. We provide a thermodynamic development of the Darcy's theory, which is consistent with the interlacement between thermal and viscoelastic relaxation effects and diffusion phenomena. The appropriate invariant convected time derivative for the flux vector and the stress tensor is discussed. This is performed by investigating the local balance laws and entropy inequality in the spatial configuration, within the single‐fluid approach. For constant parameters, our thermomechanical setting is of Jeffreys type with two delay time parameters, and hence, in the linear/linearized version, it is strictly related to phase‐lag theories within first‐order Taylor approximations. A detailed spectral analysis is carried out for the linearized version of the model, with a scrutiny to some significant limit situations, enhancing the stabilizing effects of the dissipative and elastic mechanisms, also for retardation responses. For polymeric liquids, rheological aspects, wave propagation properties and analogies with other theories with lagging are pointed out. Copyright © 2014 John Wiley & Sons, Ltd.     

Dispersive estimates for solutions to the perturbed one‐dimensional Klein‐Gordon equation with and without a one‐gap periodic potential

The knowledge about the stability properties of spatially localized structures in linear periodic media with and without defects is fundamental for many fields in nature. Its importance for the design of photonic crystals is, for example, described in 5 and 30 . Against this background, we consider a one‐dimensional linear Klein‐Gordon equation to which both a spatially periodic Lamé potential and a spatially localized perturbation are added. Given the dispersive character of the underlying equation, it is the purpose of this paper to deduce time‐decay rates for its solutions. We show that, generically, the part of the solution which is orthogonal to possible eigenfunctions of the perturbed Hill operator associated to the problem decays with a rate of w.r.t. the norm. In weighted L² norms, we even get a time decay of . Furthermore, we consider the situation of a perturbing potential that is only made up of a spatially localized part which, now, can be slightly more general. It is well‐known that, in general, it is not possible to obtain the endpoint estimate in one space dimension by means of the wave operators drawn from scattering theory. For this reason, we proceed directly and prove, along the lines of 17 , the expected decay rate of .     

Energy decay estimates and infinite blow‐up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source

This paper deals with the energy decay estimates and infinite blow‐up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source term under null Dirichlet boundary condition. By constructing a new family of potential wells, together with logarithmic Sobolev inequality and perturbation energy technique, we establish sufficient conditions to guarantee the solution exists globally or occurs infinite blow‐up and derive the polynomial or exponential energy decay estimates under some appropriate conditions.     

General decay of solutions of quasilinear wave equation with time‐varying delay in the boundary feedback and acoustic boundary conditions

In this paper, we are concerned with the general decay result of the quasi‐linear wave equation with a time‐varying delay in the boundary feedback and acoustic boundary conditions. Copyright © 2017 John Wiley & Sons, Ltd.     

Multiplicity and stability of standing waves for the nonlinear Schrödinger‐Poisson equation with a harmonic potential

In this paper, we study the multiplicity, stability, and quantitative properties of normalized standing waves for the nonlinear Schrödinger‐Poisson equation with a harmonic potential.     

Existence,blow‐up,and exponential decay estimates for a system of semilinear wave equations associated with the helicalflows of Maxwell fluid

The paper is devoted to the study of a system of semilinear wave equations associated with the helical flows of Maxwell fluid. First, based on Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we establish two local existence theorems of weak solutions. Next, we prove that any weak solutions with negative initial energy will blow up in finite time. Finally, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions via the construction of a suitable Lyapunov functional. Copyright © 2015 John Wiley & Sons, Ltd.