Time Decay Of Solutions Research Articles

On the exponential decay in time of solutions to a generalized Navier–Stokes–Fourier system

We consider a non-Newtonian incompressible heat conducting fluid with a prescribed nonuniform temperature on the boundary and with the no-slip boundary condition for the velocity. We assume no external body forces. For the power-law like models with the power law index bigger than or equal to 11/5 in three dimensions, we identify a class of solutions fulfilling the entropy equality and converging to the equilibria exponentially in a proper metric. In fact, we show the existence of a Lyapunov functional for the problem. Consequently, the steady solution is nonlinearly stable and attracts all proper weak solutions.

On the time decay for an elastic problem with three porous structures

In this paper, we study the three-dimensional porous elastic problem in the case that three dissipative mechanisms act on the three porosity structures (one in each component). It is important to remark that we consider the case when the material is not centrosymmetric, and therefore, some coupling, not previously considered in the literature concerning the time decay of solutions in porous elasticity, can appear in the system of field equations. The new couplings provided in this situation show a strong relationship between the elastic and the porous components of the material. In this situation, we obtain an existence and uniqueness result for the solutions to the problem using the Lumer-Phillips corollary to the Hille-Yosida theorem. Later, assuming a certain condition determining a “very strong” coupling between the material components, we can use the well-known arguments for dissipative semigroups to prove the exponential stability of the solutions to the problem. It is worth emphasizing that the proposed condition allows bringing the decay of the dissipative porous structure of the problem to the macroscopic elastic structure.

Pointwise decay of solutions to the energy critical nonlinear Schrödinger equations

In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schrödinger equations assuming data in L1∩H3. The main ingredients are the boundness of the Schrödinger propagators in Hardy space due to Miyachi [9] and a fractional Leibniz rule in the Hardy space. We also extend the fractional chain rule to the Hardy space.

Large Time Decay of Solutions to a Linear Nonautonomous System in Exterior Domains

In this expository paper, we study Lq-Lr decay estimates of the evolution operator generated by a perturbed Stokes system in n-dimensional exterior domains when the coefficients are time-dependent and can be unbounded at spatial infinity. By following the approach developed by the present author for the physically relevant case where the rigid motion of the obstacle is time-dependent, we clarify that some decay properties of solutions to the same system in whole space Rn together with the energy relation imply the desired estimates in exterior domains provided n≥3.

The Vlasov–Maxwell–Fokker–Planck system near Maxwellians in ℝ3

ABSTRACT Motivated by Duan and Strain [Optimal large-time behavior of the Vlasov–Maxwell–Boltzmann system in the whole space. Comm Pure Appl Math. 2011;64:1497–1546], we prove the global existence of solutions to the Vlasov–Maxwell–Fokker–Planck system near Maxwellians in the whole space by utilizing the refined energy method. In the proof of the global existence, the a priori estimates on the macroscopic component and the electromagnetic field are obtained by use of the macroscopic balance laws and the Maxwell equations, respectively. Moreover, the optimal time decay of solutions to the global Maxwellians, which can be obtained by employing the Fourier analysis.

Global well-posedness and symmetries for dissipative active scalar equations with positive-order couplings

We consider a family of dissipative active scalar equations outside the $L^{2}$-space. This was introduced in [7] and its velocity fields are coupled with the active scalar via a class of multiplier operators which morally behave as derivatives of positive order. We prove global well-posedness and time-decay of solutions, without smallness assumptions, for initial data belonging to the critical Lebesgue space $L^{\frac{n}{2\gamma-\beta}}(\mathbb{R}^{n})$ which is a class larger than that of the above reference. Symmetry properties of solutions are investigated depending on the symmetry of initial data and coupling operators.

Decay estimates for time-fractional and other non-local in time subdiffusion equations in $$\mathbb {R}^d$$ R d

We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in $$\mathbb {R}^d$$ . An important special case is the time-fractional diffusion equation, which has seen much interest during the last years, mostly due to its applications in the modeling of anomalous diffusion processes. We follow three different approaches and techniques to study this particular case: (A) estimates based on the fundamental solution and Young’s inequality, (B) Fourier multiplier methods, and (C) the energy method. It turns out that the decay behaviour is markedly different from the heat equation case, in particular there occurs a critical dimension phenomenon. The general subdiffusion case is treated by method (B) and relies on a careful estimation of the underlying relaxation function. Several examples of kernels, including the ultraslow diffusion case, illustrate our results.

On the time decay of solutions in micropolar viscoelasticity

This paper deals with isotropic micropolar viscoelastic materials. It can be said that that kind of materials have two internal structures: the macrostructure, where the elasticity effects are noticed, and the microstructure, where the polarity of the material points allows them to rotate. We introduce, step by step, dissipation mechanisms in both structures, obtain the corresponding system of equations and determine the behavior of its solutions with respect the time.

Optimal Decay Estimates for Time-Fractional and Other NonLocal Subdiffusion Equations via Energy Methods

We prove sharp estimates for the decay in time of solutions to a rather general class of nonlocal in time subdiffusion equations on a bounded domain subject to a homogeneous Dirichlet boundary condition. Important special cases are the time-fractional and ultraslow diffusion equation, which have seen much interest during the last years, mostly due to their applications in the modeling of anomalous diffusion. We study the case where the equation is in divergence form with bounded measurable coefficients. Our proofs rely on energy estimates and make use of a new and powerful inequality for integro-differential operators of the form $\partial_t (k\ast \cdot)$. The results can be generalized to certain quasilinear equations. We illustrate this by looking at the time-fractional $p$-Laplace and porous medium equation. Here, it turns out that the decay behavior is markedly different from that in the classical parabolic case.

Green’s function of a chemotaxis model in the half space and its applications

In this paper, we are interested in a degenerate Keller–Segel model in the half space xn>lt in high dimensions, with a constant l. Under the conservative boundary condition, we study the global solvability, regularity and long time behavior of solutions. We first construct Green’s function of the half space problem and study its properties. Then by Green’s function, we give the implicit formula of solutions. We prove that when the initial data is small enough, the half space problem is always globally and classically solvable. We also obtain decay estimates of solutions when the parameter l have different signs. For l<0, the time decay of solutions is (1+t)−n/2. While for l>0, the time decay is (1+t)−(n−1)/2, and in this case, we also obtain an exponential decay in the spatial space in the xn-direction.

Stability of non-constant equilibrium solutions for Euler–Maxwell equations

We consider a periodic problem for compressible Euler–Maxwell equations arising in the modeling of magnetized plasmas. The equations are quasilinear hyperbolic and partially dissipative. It is proved that smooth solutions exist globally in time and converge toward non-constant equilibrium states as the time goes to infinity. This result is obtained for initial data close to the equilibrium states with zero velocity. The proof is based on an induction argument on the order of the derivatives of solutions in energy and time dissipation estimates. We also show the global stability with exponential decay in time of solutions near the equilibrium states for compressible Euler–Poisson equations.

On estimates of time decay of solutions of one magnetic field equation in an unbounded medium

We have considered the initial boundary-value problem for one magnetic field equation in an unbounded domain with noncompact boundary. We have established estimates of the decay of solutions in L p , which depend on the geometry of this domain.

Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation

We consider the Klein–Gordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential that is constant but different on each branch. The corresponding spatial operator is self-adjoint, and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier-type inversion formula in terms of an expansion in generalized eigenfunctions. Further, we prove the surjectivity of the associated transformation, thus showing that it is in fact a spectral representation. The characteristics of the problem are marked by the non-manifold character of the star-shaped domain. Therefore, the approach via the Sturm–Liouville theory for systems is not well-suited. The considerable effort to construct explicit formulas involving the generalized eigenfunctions that incorporate the tunnel effect is justified for example by the perspective to study the influence of this effect on the L∞-time decay of solutions.

Weighted Sobolev L2 estimates for a class of Fourier integral operators

In this paper we develop elements of the global calculus of Fourier integral operators in under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev L2 estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of weighted estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.

On the Decay in Time of Solutions of the Generalized Regularized Boussinesq System

Abstract We are interested in dispersive properties of the Boussinesq system for small initial data. We prove that, for a suficiently high order nonlinearity, the solution of the Boussinesq system exists for all time and tends to zero as the time tends to infinity.

On the time decay of solutions in porous-thermo-elasticity of type II

In this paper we investigate the asymptotic behaviour of thesolutions of the linear theory of thermo-porous-elasticity. That is,we consider the theory of elastic materials with voids when the heatconduction is of type II. We assume that the only dissipationmechanism is the porous dissipation. First we prove that,generically, the solutions are exponentially stable on time or, inother words, the decay of solutions can be controlled by a negative exponential for a generic class of materials. The reason lies in the fact that the temperature is strongly coupled with both the microscopic and macroscopic structures of the materials and plays the roleof a 'driving belt' between the dissipation at the microscopicstructure and the macroscopic one. Later we note that the decay ofsolutions cannot be fast enough to make the solutions be zero in afinite period of time. Finally, we show that when the coupling termbetween the microscopic (or macroscopic) structure and the thermalvariable vanishes, the solutions do not decay exponentially(generically).

On the decay in time of solutions of some generalized regularized long waves equations

We consider the generalized Benjamin-Ono equation, regularized in the same manner that the Benjamin-Bona-Mahony equation is found from the Korteweg-de Vries equation [3], namely the equation $u_t + u_x +u^\rho u_x + H(u_{x t})=0,$ where $H$ is the Hilbert transform. In a second time, we consider the generalized Kadomtsev-Petviashvili-II equation, also regularized, namely the equation $u_t + u_x +u^\rho u_x - u_{x x t} +\partial_x^{-1}u_{y y} =0$. We are interested in dispersive properties of these equations for small initial data. We will show that, if the power $\rho$ of the nonlinearity is higher than $3$, the respective solution of these equations tends to zero when time rises with a decay rate of order close to $\frac{1}{2}$.

On the time decay of solutions in porous-elasticity with quasi-static microvoids

In this paper we investigate the temporal asymptotic behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation when the motion of microvoids is assumed to be quasi-static. This question has been recently studied in the general dynamical case. Thus, the natural question is to know if the assumption of quasi-static motion for the microvoids implies significant differences in the behavior of the solutions from the results obtained in the general dynamical case. It is worth noting that this assumption involves a qualitative change in the system of equations to be analyzed because it arises from the combination of a parabolic equation with an hyperbolic one, rather different from the well-known system of the thermo-elastic problem. First, we study the coupling of elasticity with porosity and we show that if only porous dissipation is present, the decay of solutions is slow, but if viscoelasticity is added, then the solutions decay exponentially. After that, we introduce thermal effects in the system and we show that while temperature brings exponential stability to the solutions, microtemperature does not.

On the time decay of solutions in one-dimensional theories of porous materials

In this paper we investigate the temporal asymptotic behavior of the solutions of the one-dimensional porous-elasticity problem when several damping effects are present. We show that viscoelasticity and temperature produce slow decay in time, and the same result is obtained when the porous viscosity is combined with microtemperatures. However, when the viscoelasticity is coupled with porous damping or with microtemperatures the decay is controlled by a negative exponential.

Large time decay of solutions to Burgers type equations

We study the initial value problem for the nonlinear dissipative equations where a ∈ R n , ρ > 1, σ > 0. We prove that if and then in the case σ = (ρ − 1)/(n + 1) solutions exist globally and decay as And in the case 0 < σ < (ρ − 1)/(n + 1), σ is close to (ρ − 1)/(n + 1), solutions have estimates