Exponential Decay Estimates Research Articles

Global existence and exponential decay for a viscoelastic Petrovsky system

We investigate the initial-boundary value problem for a viscoelastic Petrovsky system of Kirchhoff-type with nonlinear source term. By applying the standard continuation principle, the existence of global solutions for this problem is proved and the exponential decay estimate of global solutions is obtained.

Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity

This paper deals with the initial boundary value problem for strongly damped semilinear wave equations with logarithmic nonlinearity utt−Δu−Δut=φp(u)log|u| in a bounded domain Ω⊂Rn. We discuss the existence, uniqueness and polynomial or exponential energy decay estimates of global weak solutions under some appropriate conditions. Moreover, we derive the finite time blow up results of weak solutions, and give the lower and upper bounds for blow-up time by the combination of the concavity method, perturbation energy method and differential–integral inequality technique.

Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions

This paper is devoted to the study of a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Based on the Faedo-Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we first establish two local existence theorems of weak solutions. By the construction of a suitable Lyapunov functional, we next prove a blow up result and a decay result of global solutions.

Local indirect stabilization of N–d system of two coupled wave equations under geometric conditions

The purpose of this note is to investigate the stabilization of a system of two wave equations coupled by velocities with only one localized damping. The main novelty in this note is that the waves are not necessarily propagating at same speed and the coupling coefficient is not assumed to be positive and small. Assume that the coupling region and the damping region intersect. We prove that our system is strongly stable without geometric conditions. We then study the energy decay rate by distinguishing two cases. The first one is when the waves propagate at the same speed. In this case, under appropriate geometric conditions, we establish an exponential energy decay estimate for usual initial data. For the other case, we first show that our system is not uniformly stable. Next, under the same geometric conditions, we establish a polynomial energy decay of type 1t for smooth initial data. Finally, in one space dimension, using the real part of the asymptotic expansion of eigenvalues of the system, we prove that the obtained polynomial decay rate is optimal.

Exponential decay estimates for fundamental solutions of Schrödinger-type operators

In the present paper, we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators L = − ( ∇ − i a ) T A ( ∇ − i a ) + V L=-(\nabla -i\mathbf {a})^TA(\nabla -i\mathbf {a})+V . The latter class includes, in particular, the magnetic Schrödinger operator − ( ∇ − i a ) 2 + V -\left (\nabla -i\mathbf {a}\right )^2+V and the generalized electric Schrödinger operator − div A ∇ + V -\textrm {div }A\nabla +V . Our exponential decay bounds rest on a generalization of the Fefferman–Phong uncertainty principle to the present context and are governed by the Agmon distance associated with the corresponding maximal function. In the presence of a scale-invariant Harnack inequality—for instance, for the generalized electric Schrödinger operator with real coefficients—we establish both lower and upper estimates for fundamental solutions, thus demonstrating the sharpness of our results. The only previously known estimates of this type pertain to the classical Schrödinger operator − Δ + V -\Delta +V .

Global stability of homogeneous steady states in scaling-invariant spaces for a Keller–Segel–Navier–Stokes system

In this paper, we study the global stability of homogeneous equilibria in Keller–Segel–Navier–Stokes equations in scaling-invariant spaces. We prove that for any given 0<M<1+μ1 with μ1 being the first eigenvalue of Neumann Laplacian, the initial–boundary value problem of the Keller–Segel–Navier–Stokes system has a unique globally bounded classical solution provided that the initial datum is chosen sufficiently close to (M,M,0) in the norm of Ld/2(Ω)×W˙1,d(Ω)×Ld(Ω) and satisfies a natural average mass condition. Our proof is based on the perturbation theory of semigroups and certain delicate exponential decay estimates for the linearized semigroup. Our result suggests a new observation that nontrivial classical solution for Keller–Segel–Navier–Stokes equation can be obtained globally starting from suitable initial data with arbitrarily large total mass provided that volume of the bounded domain is large, correspondingly.

Existence, blow-up and exponential decay estimates for the nonlinear Kirchhoff-Carrier wave equation in an annular with nonhomogeneous Dirichlet conditions

This paper is devoted to the study of a nonlinear Kirchhoff-Carrier wave equation in an annular associated with nonhomogeneous Dirichlet conditions. At first, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we prove a blow-up result for solutions with a negative initial energy and establish a sufficient condition to obtain the exponential decay of weak solutions.

Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms

In this paper we consider a porous-elastic system consisting of nonlinear boundary/interior damping and nonlinear boundary/interior sources. Our interest lies in the theoretical understanding of the existence, finite time blow-up of solutions and their exponential decay using non-trivial adaptations of well-known techniques. First, we apply the conventional Faedo-Galerkin method with standard arguments of density on the regularity of initial conditions to establish two local existence theorems of weak solutions. Moreover, we detail the uniqueness result in some specific cases. In the second theme, we prove that any weak solution possessing negative initial energy has the latent blow-up in finite time. Finally, we obtain the so-called exponential decay estimates for the global solution under the construction of a suitable Lyapunov functional. In order to corroborate our theoretical decay, a numerical example is provided.

Well‐Posedness and Global Behavior of the Peskin Problem of an Immersed Elastic Filament in Stokes Flow

AbstractWe consider the problem of a one‐dimensional elastic filament immersed in a two‐dimensional steady Stokes fluid. Immersed boundary problems in which a thin elastic structure interacts with a surrounding fluid are prevalent in science and engineering, a class of problems for which Peskin has made pioneering contributions. Using boundary integrals, we first reduce the fluid equations to an evolution equation solely for the immersed filament configuration. We then establish local well‐posedness for this equation with initial data in low‐regularity Hölder spaces. This is accomplished by first extracting the principal linear evolution by a small‐scale decomposition and then establishing precise smoothing estimates on the nonlinear remainder. Higher regularity of these solutions is established via commutator estimates with error terms generated by an explicit class of integral kernels. Furthermore, we show that the set of equilibria consists of uniformly parametrized circles and prove nonlinear stability of these equilibria with explicit exponential decay estimates, the optimality of which we verify numerically. Finally, we identify a quantity that respects the symmetries of the problem and controls global‐in‐time behavior of the system. © 2018 Wiley Periodicals, Inc.

Asymptotic stability for spectrally stable Lugiato-Lefever solitons in periodic waveguides

We consider the Lugiato-Lefever model of optical fibers in the periodic context. Spectrally stable periodic steady states were constructed recently in the studies of Delcey and Haragus [Philos. Trans. R. Soc., A 376, 20170188 (2018)]; [Rev. Roumaine Math. Pures Appl. (to be published)]; and Hakkaev et al. (e-print arXiv:1806.04821). The spectrum of the linearization around such solitons consists of simple eigenvalues 0, −2α &amp;lt; 0, while the rest of it is a subset of the vertical line {μ:Rμ=−α}. Assuming such a property abstractly, we show that the linearized operator generates a C0 semigroup and, more importantly, the semigroup obeys (optimal) exponential decay estimates. Our approach is based on the Gearhart-Prüss theorem, where the required resolvent estimates may be of independent interest. These results are applied to the proof of asymptotic stability with phase of the steady states.

Decay estimates for evolution equations with classical and fractional time-derivatives

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving both standard and Caputo time-derivative, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators.Both local and fractional space diffusion are taken into account, possibly in a nonlinear setting. The different quantitative behaviours, which distinguish polynomial decays from exponential ones, depend heavily on the structure of the time-derivative involved in the equation.

Exponential decay estimates of the eigenvalues for the Neumann-Poincare operator on analytic boundaries in two dimensions

We show that the eigenvalues of the Neumann-Poincare operator on analytic boundaries of simply connected bounded planar domains tend to zero exponentially fast, and the exponential convergence rate is determined by the maximal Grauert radius of the boundary. We present a few examples of boundaries to show that the estimate is optimal.

The frequency and the structure of large character sums

Let M(\chi) denote the maximum of |\sum_{n\le N}\chi(n)| for a given non-principal Dirichlet character \chi modulo q , and let N_\chi denote a point at which the maximum is attained. In this article we study the distribution of M(\chi)/\sqrt{q} as one varies over characters modulo q , where q is prime, and investigate the location of N_\chi . We show that the distribution of M(\chi)/\sqrt{q} converges weakly to a universal distribution \Phi , uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for \Phi 's tail. Almost all \chi for which M(\chi) is large are odd characters that are 1-pretentious. Now, M(\chi)\ge |\sum_{n\le q/2}\chi(n)| = \frac{|2-\chi(2)|}\pi \sqrt{q} |L(1,\chi)| , and one knows how often the latter expression is large, which has been how earlier lower bounds on \Phi were mostly proved. We show, though, that for most \chi with M(\chi) large, N_\chi is bounded away from q/2 , and the value of M(\chi) is little bit larger than \frac{\sqrt{q}}{\pi} |L(1,\chi)| .

On high-order approximations for describing the lagging behavior of heat conduction

This paper is dedicated to high-order effects of thermal lagging in correlation with heat transfer models in micro- or nanoscale, relating the number of energy carriers and the associated resonance phenomenon under high-frequency excitations. Thus, a class of constitutive equations is considered for the heat flux describing high-order effects in the lagging behavior of heat transport. Tzou’s model, which is based on time-differential dual-phase-lag approximations of heat conduction, is generalized, incorporating the microstructural interaction effect in the fast-transient process of heat transport. More precisely, polynomial approximations of order n for the heat flux vector and of order m for the gradient of the temperature variation are considered. Further, well-posedness is established for solutions of the specific initial boundary value problems for the mathematical model when: (i) ; (ii) ; and (iii) . This means that uniqueness and continuous dependences of solutions are established with respect to the given prescribed data. With this aim, some modified initial boundary value problems related to the operators of the constitutive equation are introduced and suitable time-weighted measures of the solution are presented, and are used to establish the uniqueness theorems and to generate some estimates describing the continuous dependence of solutions with respect to the given data for each of the three cases specified. Moreover, the spatial behavior of the transient solutions is studied and an influence domain result is established for , while for some exponential decay estimates of Saint-Venant type are established. All results are established without restrictions on the characteristic parameters of the constitutive equation other than that the product of the coefficients of the time derivatives of greatest order of the two operators involved in the constitutive equation is positive.

Well-posedness and stability results for nonlinear abstract evolution equations with time delays

We consider abstract evolution equations with a nonlinear term depending on the state and on delayed states. We show that, if the $$C_0$$ -semigroup describing the linear part of the model is exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions on the nonlinearity. More precisely, we give a general exponential decay estimate for small time delays if the nonlinear term is globally Lipschitz and an exponential decay estimate for solutions starting from small data when the nonlinearity is only locally Lipschitz and the linear part is a negative selfadjoint operator. In the latter case we do not need any restriction on the size of the time delays. In both cases, concrete examples are presented that illustrate our abstract results.

Spatial Behavior of the Steady State Vibrations in a Dual-Phase-Lag Rigid Conductor

This paper studies the spatial behavior of the steady state vibrations in a cylinder made of a dual-phase-lag anisotropic rigid conductor material. We analyze the influence of the lagging model upon the spatial behavior of the amplitude of vibration along the axis of the cylinder, providing the explicit expressions of the decay rate and of the corresponding critical frequency in terms of the coefficients of the considered constitutive equation (or delay times). In fact, for the amplitude of the harmonic vibrations we obtain some exponential decay estimates of Saint-Venant type, provided the frequency of vibration is lower than a critical value. This gives information on the thermal penetration depth of the steady state vibrations describing the heat affected zone. Illustrative examples are given for the class of lagging behavior models that are thermodynamically compatible.

Asymptotic Analysis and Optimal Error estimates for Benjamin‐Bona‐Mahony‐Burgers' Type Equations

In this article, stabilization result for the Benjamin‐Bona‐Mahony‐Burgers' (BBM‐B) equation, that is, convergence of unsteady solution to steady state solution is established under the assumption that a linearized steady state eigenvalue problem has a minimal positive eigenvalue. Based on appropriate conditions on the forcing function, exponential decay estimates in , and ‐norms are derived, which are valid uniformly with respect to the coefficient of dispersion as it tends to zero. It is, further, observed that the decay rate for the BBM‐B equation is smaller than that of the decay rate for the Burgers equation. Then, a semidiscrete Galerkin method for spatial direction keeping time variable continuous is considered and stabilization results are discussed for the semidiscrete problem. Moreover, optimal error estimates in ‐norms preserving exponential decay property are established using the steady state error estimates. For a complete discrete scheme, a backward Euler method is applied for the time discretization and stabilization results are again proved for the fully discrete problem. Subsequently, numerical experiments are conducted, which verify our theoretical results. The article is finally concluded with a brief discussion on an extension to a multidimensional nonlinear Sobolev equation with Burgers' type nonlinearity.

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.

A general decay result for a memory-type Timoshenko-thermoelasticity system with second sound

In this paper we consider a system of Timoshenko-thermoelasticity with second sound in the presence of an infinite history term. We prove, under suitable conditions on the coefficients and the relaxation function, a general decay result from which the exponential and polynomial decay estimates are only special cases. We also establish a lack of exponential decay result and discuss the optimality for some situation.

Qualitative properties of solutions in the time differential dual-phase-lag model of heat conduction

In this paper we study the time differential dual-phase-lag model of heat conduction incorporating the microstructural interaction effect in the fast-transient process of heat transport. We analyse the influence of the delay times upon some qualitative properties of the solutions of the initial boundary value problems associated to such a model. Thus, the uniqueness results are established under the assumption that the conductivity tensor is positive definite and the delay times $\tau_q$ and $\tau_T$ vary in the set $\{0\leq \tau_q\leq 2\tau_T\}\cup \{0<2\tau_T< \tau_q\}$. For the continuous dependence problem we establish two different estimates. The first one is obtained for the delay times with $0\leq \tau_q \leq 2\tau_T$, which agrees with the thermodynamic restrictions on the model in concern, and the solutions are stable. The second estimate is established for the delay times with $0<2\tau_T< \tau_q$ and it allows the solutions to have an exponential growth in time. The spatial behavior of the transient solutions and the steady-state vibrations is also addressed. For the transient solutions we establish a theorem of influence domain, under the assumption that the delay times are in $\left\{0<\tau_q\leq 2\tau_T\right\}\cup \left\{0<2\tau_T<\tau_q\right\}$. While for the amplitude of the harmonic vibrations we obtain an exponential decay estimate of Saint-Venant type, provided the frequency of vibration is lower than a critical value and without any restrictions upon the delay times.