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We consider an abstract linear wave equation with a time-dependent dissipation that decays at infinity with the so-called scale invariant rate, which represents the critical case. We do not assume that the coefficient of the dissipation term is smooth, and we investigate the effect of its oscillations on the decay rate of solutions. We prove a decay estimate that holds true regardless of the oscillations. Then we show that oscillations that are too fast have no effect on the decay rate, while oscillations that are in resonance with one of the frequencies of the elastic part can alter the decay rate. In the proof we first reduce ourselves to estimating the decay of solutions to a family of ordinary differential equations, then by using polar coordinates we obtain explicit formulae for the energy decay of these solutions, so that in the end the problem is reduced to the analysis of the asymptotic behavior of suitable oscillating integrals. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of planar viscous shock wave for the 3-dimensional compressible Navier-Stokes-Poisson (NSP) system in the domain Ω : = R × T 2 with T 2 = (R / Z) 2. The stability problem of viscous shock under small 1-dimensional perturbations was solved in Duan-Liu-Zhang [7]. In this paper, we prove the viscous shock is still stable under small 3-d perturbations. Firstly, we decompose the perturbation into the zero mode and non-zero mode. Then we can show that both the perturbation and zero-mode time-asymptotically tend to zero by the anti-derivative technique and crucial estimates on the zero-mode. Moreover, we can further prove that the non-zero mode tends to zero with exponential decay rate. The key point is to estimate the non-zero mode of nonlinear terms involving electronic potential, see Lemma 6.1 below. [ABSTRACT FROM AUTHOR]

We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the L ∞ norm of these solutions. [ABSTRACT FROM AUTHOR]

Consider the non-stationary Navier-Stokes equation in the upper half space of R n. We prove the decay estimates of the strong solution and its derivatives in the setting of Besov and Triebel–Lizorkin spaces. This is the first time that the decay estimates of solutions to the non-stationary Navier-Stokes equations on Besov and Triebel-Lizorkin spaces are established. Our results not only extend the known results from Hardy spaces to Besov and Triebel-Lizorkin spaces but also imply new estimates on Sobolev spaces and give better decay estimates on Hardy spaces. [ABSTRACT FROM AUTHOR]

In this work, we investigate the global well-posedness and asymptotic behavior of a mathematical model of ultrasound-induced heating based on a coupled system of Westervelt's nonlinear acoustic wave equation and Pennes bioheat equation. To this end, under Dirichlet–Dirichlet boundary conditions, we prove global existence for sufficiently small and smooth solutions of the nonlinear model using an energy method. In addition, we show that the energy norm of the resulting pressure and temperature decays to the steady state exponentially fast. [ABSTRACT FROM AUTHOR]

We revisit the classical work of Huang-Matsumura-Xin [9] for contact wave of the one-dimensional compressible Navier-Stokes system under the zero mass condition. The large-time asymptotic stability of a contact wave pattern with a uniform convergence rate (1 + t) − 1 4 was proved by Huang-Matsumura-Xin. In this paper, Huang-Matsumura-Xin's convergence rate is improved to (1 + t) − 5 8 ln 1 2 ⁡ (2 + t) by using a new Poincaré type inequality and a detailed energy analysis. Our work is motivated by a problem arising in [9]. [ABSTRACT FROM AUTHOR]

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

Navier–Stokes system on the half space with periodically oscillating boundary has a time periodic solution which depends on time variable and vertical variable only. It is proved that the time periodic solution is asymptotically stable when the Reynolds number is sufficiently small; and the decay estimates of the perturbations are established in the frameworks of both strong and weak solutions. [ABSTRACT FROM AUTHOR]

We study a temporally third order (Moore–Gibson–Thompson) equation with a memory term. Previously it was known that, in non-critical regime, the global solutions exist and the energy functionals decay to zero. More precisely, it is known that the energy has exponential decay if the memory kernel decays exponentially. The current work is a generalization of the previous one (Part I) in that it allows the memory kernel to be more general and shows that the energy decays the same way as the memory kernel does, exponentially or not. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Cauchy problem for the one-dimensional Timoshenko system coupled with heat conduction, wherein the latter is described by either the Cattaneo law or the Fourier law. We prove that heat dissipation alone is sufficient to stabilize the system in both cases, so that additional mechanical damping is unnecessary. However, the decay of solutions without the mechanical damping is found to be slower than that with mechanical damping. Furthermore, in contrast to earlier results of Said-Houari and Kasimov (2012) [10] and Fernández Sare and Racke (2009) [12], we find that the Timoshenko–Fourier and the Timoshenko–Cattaneo systems have the same decay rate. The rate depends on a certain number α (first identified by Santos et al., 2012 [11] in a related study in a bounded domain), which is a function of the parameters of the system. [Copyright &y& Elsevier]

Abstract: Using the Stokes solution formula and – estimates of the Stokes operator semigroup, we establish the weighted decay properties for the Stokes flow and Navier–Stokes equations including their spatial derivatives in half spaces. In addition, the unboundedness of the projection operator is overcome by employing a decomposition for the nonlinear term, and -asymptotic behavior for the second derivatives of Navier–Stokes flows in half spaces is given. [Copyright &y& Elsevier]

Abstract: The algebraic decay rates for the total kinetic energy of weak solutions of the n-dimensional viscous Boussinesq system in exterior domains are established by means of the spectral decomposition method of fractional powers of the Laplacian and Stokes operators. The results reveal that -decay of the velocity field with zero divergence becomes slower than that of the temperature. [Copyright &y& Elsevier]

Abstract: In this paper, we use the potential analysis to study the properties of the positive solutions of a γ-Laplace system in Here , satisfy the critical condition . First, the positive solutions u and v satisfy an integral system involving the Wolff potentials. We then use the method of regularity lifting to obtain an optimal integrability for this Wolff type integral system. Different from the case of , it is more difficult to handle the asymptotics since u and v have not radial structures. We overcome this difficulty by a new method and obtain the decay rates of u and v as . We believe that this new method is appropriate to deal with the asymptotics of other decaying solutions without the radial structures. [Copyright &y& Elsevier]

Abstract: In a half space, we consider the asymptotic behavior of the strong solution for the non-stationary Navier–Stokes equations. In particular, the decay rates of the second order derivatives of the Navier–Stokes flows in () with are derived by using estimates and a clever analysis on the fractional powers of the Stokes operator. In addition, we prove that the strong solution and its first and second derivatives decay in time more rapidly than observed in general if the initial datum lies in a suitable weighted space. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the damped wave equation with space–time dependent potential and absorbing semilinear term . Here, with , and . Based on the local existence theorem, we obtain the global existence and the decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the initial–boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation in the half-space Here is an unknown function of and , are two given constant states and the nonlinear function is assumed to be a strictly convex function of u. We first show that the corresponding boundary layer solution of the above initial–boundary value problem is global nonlinear stable and then, by employing the space–time weighted energy method which was initiated by Kawashima and Matsumura [S. Kawashima, A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985) 97–127], the convergence rates (both algebraic and exponential) of the global solution to the above initial–boundary value problem toward the boundary layer solution are also obtained for both the non-degenerate case and the degenerate case . [Copyright &y& Elsevier]

Abstract: We construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder, which is close to a uniform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infinite nozzle a little bit, we can obtain a transonic wave near the uniform one. As a consequence, we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall. Based on the theory of [G.Q. Chen, M. Feldman, Existence and stability of multi-dimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007) 185–242], the crucial parts of this paper are to derive the uniform Schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder. [Copyright &y& Elsevier]

Abstract: We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation on a bounded domain, subject to Dirichlet boundary and to initial conditions. The data are supposed to satisfy suitable regularity and growth conditions. Our approach to the convergence result and decay estimate is based on the Łojasiewicz–Simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates. The abstract results and their applications are discussed also in the framework of Orlicz–Sobolev spaces. [Copyright &y& Elsevier]

Abstract: We study the problem of decay rate for the solutions of the initial–boundary value problem to the wave equation, governed by localized nonlinear dissipation and without any assumption on the dynamics (i.e., the control geometric condition is not satisfied). We treat separately the autonomous and the non-autonomous cases. Providing regular initial data, without any assumption on an observation subdomain, we prove that the energy decays at last, as fast as the logarithm of time. Our result is a generalization of Lebeau (in: A. Boutet de Monvel, V. Marchenko (Eds.), Algebraic and Geometric Methods in Mathematical Physics, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1996, pp. 73) result in the autonomous case and Nakao (Adv. Math. Sci. Appl. 7 (1) (1997) 317) work in the non-autonomous case. In order to prove that result we use a new method based on the Fourier–Bross–Iaglintzer (FBI) transform. [Copyright &y& Elsevier]

We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition and study the asymptotic behavior of the solution for large times. For possible application we are interested to show a decay estimate which does not involve weighted norms of the initial data. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution. [Copyright &y& Elsevier]