Decay Of Solutions Research Articles

Stabilization for degenerate equations with drift and small singular term

We consider a degenerate/singular wave equation in lone dimension, with drift and in presence of a leading operator that is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

Results on a nonlinear wave equation with acoustic and fractional boundary conditions coupling by logarithmic source and delay terms: Global existence and Asymptotic behavior of solutions

The nonlinear wave equation with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, is notable for its capacity to model complex systems, contribute to the advancement of mathematical theory, and exhibit wide-ranging applicability to real-world problems. This paper investigates the global existence and general decay of solutions to a wave equation characterized by the inclusion of logarithmic source and delay terms, governed by both fractional and acoustic boundary conditions. The global existence of solutions is analyzed under various hypotheses, and the general decay behavior is established through the construction and application of a suitable Lyapunov function

Structural stability for double-diffusion Brinkman fluid with Soret effect

This paper considers the double diffusive Brinkman flow in a semi-infinite pipe. By establishing a priori estimates of the solutions and setting an appropriate “energy” function, we not only obtain the continuous dependence and convergence of the solution on the Soret coefficient but also prove that the solutions decay exponentially with the distance from the finite end of the cylinder.

Hydrodynamic limit for the non-cutoff Boltzmann equation

This work deals with the non-cutoff Boltzmann equation for all types of potentials, in both the torus \mathbf{T}^{3} and in the whole space \mathbf{R}^{3} , under the incompressible Navier–Stokes scaling. We first establish the well-posedness and decay of global mild solutions to this rescaled Boltzmann equation in a perturbative framework, that is, for solutions close to the Maxwellian, obtaining in particular integrated-in-time regularization estimates. We then combine these estimates with spectral-type estimates in order to obtain the strong convergence of solutions to the non-cutoff Boltzmann equation towards the incompressible Navier–Stokes–Fourier system.

Global existence and exponential decay of strong solutions to the 3D nonhomogeneous nematic liquid crystal flows with density-dependent viscosity

Global existence and exponential decay of strong solutions to the 3D nonhomogeneous nematic liquid crystal flows with density-dependent viscosity

The optimal polynomial decay in the extensible Timoshenko system

AbstractIn this paper, we derive the equations that constitute the nonlinear mathematical model of an extensible thermoelastic Timoshenko system. The nonlinear governing equations are derived by applying the Hamilton principle to full von Kármán equations. The model takes account of the effects of extensibility, where the dissipations are entirely contributed by temperature. Based on the semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By using a resolvent criterion, developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the considered problem under the condition (65). Moreover, by an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we show the non‐exponential stability of the same problem; but strongly stable by following a result due to Arendt–Batty. In the absence of additional mechanical dissipations, the system is often not highly stable. By adding a damping frictional function to the first equation of the nonlinear derived model with extensibility and using the multiplier method, we show that the solutions decay exponentially if Equation (85) holds.

Results from a Nonlinear Wave Equation with Acoustic and Fractional Boundary Conditions Coupling by Logarithmic Source and Delay Terms: Global Existence and Asymptotic Behavior of Solutions

The nonlinear wave equation with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, is significant for its ability to model complex systems, its contribution to the advancement of mathematical theory, and its wide-ranging applicability to real-world problems. This paper examines the global existence and general decay of solutions to a wave equation characterized by coupling with logarithmic source and delay terms, and governed by both fractional and acoustic boundary conditions. The global existence of solutions is analyzed under a range of hypotheses, and the general decay behavior is established through the construction and application of an appropriate Lyapunov function.

Global well-posedness for the three dimensional compressible micropolar equations

In this paper, we study the Cauchy problem of the three-dimensional compressible micropolar equations in the absence of heat-conductivity. By leveraging Fourier theory and employing a refined energy method, we establish the global well-posedness of the equations for small initial data within Besov spaces. As a byproduct, we also derive the optimal time decay of solutions if the low frequency of initial data belonging to Ḃ2,∞−σ1(R3).

Qualitative properties for a Moore–Gibson–Thompson thermoelastic problem with heat radiation

In this work, we study some qualitative properties arising in the solution of a thermoelastic problem with heat radiation. The so-called Moore–Gibson–Thompson equation is used to model the heat conduction. By using the logarithmic convexity argument, the uniqueness and instability of solutions are proved without imposing any condition on the elasticity tensor. Then, the existence of solutions is obtained assuming that the elastic tensor is positive definite applying the theory of linear semigroups, and the exponential energy decay is shown in the one-dimensional case. Finally, we consider the one-dimensional quasi-static version and we assume that the elastic coefficient is negative. The existence and decay of solutions are proved, and a justification of the quasi-static approach is also provided.

On a Viscoelastic Plate Equation with Logarithmic Nonlinearity and Variable-Exponents: Global Existence, General Decay and Blow-Up of Solutions

On a Viscoelastic Plate Equation with Logarithmic Nonlinearity and Variable-Exponents: Global Existence, General Decay and Blow-Up of Solutions

Global Well-Posedness and Exponential Decay of Strong Solution to the Three-Dimensional Nonhomogeneous Bénard System with Density-Dependent Viscosity and Vacuum

Global Well-Posedness and Exponential Decay of Strong Solution to the Three-Dimensional Nonhomogeneous Bénard System with Density-Dependent Viscosity and Vacuum

On Estimation of the Decay of Solutions of an Initial-Boundary-Value Problem for a System of Semilinear Equations of Magnetoelasticity with Dissipative Term in Exterior Domains

On Estimation of the Decay of Solutions of an Initial-Boundary-Value Problem for a System of Semilinear Equations of Magnetoelasticity with Dissipative Term in Exterior Domains

On $$L^2$$ decay of weak solutions of several incompressible fluid models

On $$L^2$$ decay of weak solutions of several incompressible fluid models

Global well-posedness and decay to 3D MHD equations with nonlinear damping

In this paper, we consider the Cauchy problem for the 3D MHD system with nonlinear damping terms κ|u|α−1u and γ|b|β−1b and prove the existence and uniqueness of global strong solution when one of the following two conditions is verified (1)3≤α<4,β≥6α−1+1;(2)α≥4,β≥1. This greatly improves the previous results. Moreover, we also discuss the uniqueness of weak solutions when α and β satisfy certain conditions. Finally, we study the decay of weak solutions when α,β>73 and obtain the upper bound for the derivative of strong solution when 3≤α<4,β≥6α−1+1 or α≥4,β>73.

Feedback boundary control of multi-dimensional hyperbolic systems with relaxation

This paper is concerned with boundary stabilization of multi-dimensional hyperbolic systems of partial differential equations. By adapting the Lyapunov function previously proposed by the second author for linearized hyperbolic systems with relaxation structure, we derive certain control laws so that the corresponding solutions decay exponentially in time. The result is illustrated with an application to water flows in open channels. The effectiveness of the derived control laws is confirmed by numerical simulations.

On decay of solutions to some perturbations of the Korteweg-de Vries equation

On decay of solutions to some perturbations of the Korteweg-de Vries equation

Singlet oxygen generation proportion from triplet state of porphyrin in water

The proportion of singlet oxygen (R) produced from the triplet state of porphyrin in aqueous solutions was investigated by substituting porphyrin molecules with the metalloporphyrin Platinum(II) meso-tetraphenylporphyrin (Pt-TPP). The feasibility of using platinum porphyrin as a substitute for porphyrin to study the photophysical properties in photodynamic therapy (PDT) was confirmed. The model of first-order decay in a water-methanol mixed solution encompasses potential deactivation processes of the excited triplet states and was built, which was confirmed by monitoring the change in the total transition rate of the T1 state (A0) value at different water contents under anoxic environments, obtained through measuring the phosphorescence lifetime (τp) of Pt-TPP. Combined with A0 and the quenching rate constant of T1 by O2 (kq), the formula for R in relation to changes in oxygen can be deduced in pure water.

Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: Application to the nonlinearly damped $p$-system

A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta–Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for low ones. This allows us to derive a sharp description of the space-time decay of solutions for large time. However, such analysis relies heavily on the use of the Fourier transform, which we avoid here, developing the “physical space version” of the hyperbolic hypocoercivity approach introduced in Beauchard and Zuazua [Arch. Ration. Mech. Anal. 199 (2011), 177–227], to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows us to recover the natural heat-like time decay of solutions under sharp rank conditions, without employing Fourier analysis or L^{1} assumptions on the initial data. Taking advantage of this Fourier-free framework, we establish new enhanced time-decay estimates for initial data belonging to weighted Sobolev spaces. These results are then applied to the nonlinear compressible Euler equations with linear damping. We also prove the logarithmic stability of the nonlinearly damped p -system.

Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem

Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem

Time-Velocity Decay of Solutions to the Non-cutoff Boltzmann Equation in the Whole Space

Time-Velocity Decay of Solutions to the Non-cutoff Boltzmann Equation in the Whole Space