Decay Of Solutions Research Articles

Normal trace inequalities and decay of solutions to the nonlinear Maxwell system with absorbing boundary

We study the quasilinear Maxwell system with a strictly positive, state dependent boundary conductivity. For small data we show that the solution exists for all times and decays exponentially to 0. As in related literature we assume a nontrapping condition. Our approach relies on a new trace estimate for the corresponding non-autonomous linear problem, an observability-type estimate, and a detailed regularity analysis. The results are improved in the linear autonomous case, using properties of the Helmholtz decomposition in Sobolev spaces of (small) negative order.

Well-posedness and general energy decay of solutions for a nonlinear damping piezoelectric beams system with thermal and magnetic effects

In this article, we study the piezoelectric beams with thermal and magnetic effects in the presence of a nonlinear damping term acting on the mechanical equation. First, we prove that the system is well-posed in the sense of semigroup theory. And by constructing a suitable Liapunov functional, we show a general decay result of the solution for the system from which the polynomial and exponential decay are only special cases. Furthermore, our result does not depend on any relationship between system parameters.

Coupled system of nonlinear viscoelastic plate equations of (p(x),q(x))‐Kirchhoff‐type: Global existence, general decay, and blow‐up

This work studies the global behavior of solutions for a system of ‐Kirchhoff‐type equations. We prove the global existence of solutions, and next, by constructing suitable auxiliary functionals, the general decay of solutions has been confirmed when the exponents satisfy appropriate conditions. Also, under suitable conditions on data, we establish the finite time blow‐up of solutions with negative initial energy. Our results extend and improve many earlier results in the literature.

Non–uniform decay of solutions to the incompressible magneto–micropolar fluids with/without magnetic diffusion and spin viscosity

We are concerned with the long time behavior of solutions to the incompressible magneto–micropolar fluids with/without magnetic diffusion and spin viscosity in two and three dimensions. When the magnetic diffusion is absent, we show that L2–norm of fluid velocity and micro–rotational velocity vanishes and L2–norm of the magnetic field converges to a non–negative constant as time tends to infinity. Moreover, when both magnetic diffusion and spin viscosity are present and the initial data belongs to L2, we prove that the solutions decay to zero without a rate, and this non–uniform decay is optimal. The proofs are based on Fourier splitting method, low–frequency and high–frequency decomposition techniques and delicate energy estimates.

Stability of 2D inviscid MHD equations with only vertical magnetic diffusion on T2

Physical experiments and numerical simulations have observed a remarkable phenomenon that a background magnetic field can smooth and stabilize the electrically conducting turbulent fluids. To understand the mechanism of this phenomenon, we will focus on a special 2D magnetohydrodynamic (MHD) system with no viscosity and partial magnetic resistive and examine the stability near a background magnetic field. Due to the lack of dissipation for velocity field, this stability problem is not trivial. Without the presence of a magnetic field, the fluid velocity is governed by the 2D incompressible Euler equation. And it is well known that solutions to the 2D incompressible Euler equation can grow rather rapidly. Our result in this paper then shows the stabilization effect of magnetic filed for conductive fluids. By coupling a suitable magnetic filed, we can obtain the uniform upper bound of solutions. Moreover, we will derive the exponentially decay of solutions on one direction. The approach is based on delicate energy estimate together with some observations such as Poincáre’s inequality in one direction and cancellation structure.

Existence and decay of solutions to coupled systems of nonlinear wave equations with variable exponents

In this article, we consider a coupled system of two hyperbolic equations with variable exponents in the damping and source terms, where the dampings are modilated with time-dependent coefficients \(\alpha(t), \beta(t)\). First, we state and prove an existence result of a global weak solution, using Galerkin's method with compactness arguments. Then, by a Lemma due to Martinez, we establish the decay rates of the solution energy, under suitable assumptions on the variable exponents \(m\) and \(r\) and the coefficients \( \alpha\) and \(\beta\). To illustrate our theoretical results, we give some numerical examples.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/73/abstr.html

Existence and General Decay of Solutions for a Weakly Coupled System of Viscoelastic Kirchhoff Plate and Wave Equations

In this paper, a weakly coupled system (by the displacement of symmetric type) consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain is considered. Under the assumptions on a more general type of relaxation functions, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions.

Initial boundary value problem for a viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term: decay estimates and blow-up result

In this paper, we study the initial boundary value problem for the following viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term where the relaxation function satisfies g'(t)leq -xi (t)g^{r}(t), tgeq 0, 1leq r< frac{3}{2}. The main goal of this work is to study the global existence, general decay, and blow-up result. The global existence has been obtained by potential-well theory, the decay of solutions of energy has been established by introducing suitable energy and Lyapunov functionals, and a blow-up result has been obtained with negative initial energy.

Classification of Blow-up and Global Existence of Solutions to a System of Petrovsky Equations

In this paper, we investigate global existence, uniform decay, and blow-up of solutions for a class of system of Petrovsky equations containing nonlinear damping and sources. By introducing a family of potential wells, we not only obtain the invariant sets and vacuum isolating of solutions but also give some threshold results of global existence and nonexistence of solutions. Furthermore, by using energy techniques, we also establish certain qualitative estimates for solution.

General decay and blow-up of solution for a viscoelastic wave equation with nonlinear degenerate damping and source terms

We are concerned with the initial-boundary value problem for a nonlinear viscoelastic problem with the degenerate damping and the nonlinear source term in this paper. Under appropriate assumptions on the initial data, memory kernel and parameters, we establish several results regarding the global solution, blow-up properties and general decay in the case of positive initial energy. We overcome the difficulties brought by positive initial energy by using the potential well theory.

Scattering of an inhomogeneous coupled Schrödinger system in the conformal space

Abstract This paper studies the inhomogeneous defocusing coupled Schrödinger system i ⁢ u ˙ j + Δ ⁢ u j = | x | - ρ ⁢ ( ∑ 1 ≤ k ≤ m a j ⁢ k ⁢ | u k | p ) ⁢ | u j | p - 2 ⁢ u j , ρ &gt; 0 , j ∈ [ 1 , m ] . i\dot{u}_{j}+\Delta u_{j}=\lvert x\rvert^{-\rho}\bigg{(}\sum_{1\leq k\leq m}a_% {jk}\lvert u_{k}\rvert^{p}\biggr{)}\lvert u_{j}\rvert^{p-2}u_{j},\quad\rho&gt;0,% \,j\in[1,m]. The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 ⁢ ( ℝ N ) {f\in H^{1}(\mathbb{R}^{N})} such that x ⁢ f ∈ L 2 ⁢ ( ℝ N ) {xf\in L^{2}(\mathbb{R}^{N})} . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x ⁢ u 0 ∈ L 2 {xu_{0}\in L^{2}} enables us to get the scattering in the mass-sub-critical regime p 0 &lt; p ≤ 2 - ρ N + 1 {p_{0}&lt;p\leq\frac{2-\rho}{N}+1} , where p 0 {p_{0}} is the Strauss exponent. The proof is based on the decay of global solutions coupled with some non-linear estimates of the source term in Strichartz norms and some standard conformal transformations. Precisely, one gets | t | α ⁢ ∥ u ⁢ ( t ) ∥ L r ⁢ ( ℝ N ) ≲ 1 \lvert t\rvert^{\alpha}\lVert u(t)\rVert_{L^{r}(\mathbb{R}^{N})}\lesssim 1 for some α &gt; 0 {\alpha&gt;0} and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ ⁢ u 0 {e^{i\cdot\Delta}u_{0}} . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 &lt; p &lt; 2 - ρ N - 2 + 1 {p_{0}&lt;p&lt;\frac{2-\rho}{N-2}+1} . This helps to better understand the asymptotic behavior of the energy solutions. Indeed, the source term has a negligible effect for large time and the above non-linear Schrödinger problem behaves like the associated linear one. In order to avoid a singular source term, one assumes that p ≥ 2 {p\geq 2} , which restricts the space dimensions to N ≤ 3 {N\leq 3} . In a paper in progress, the authors treat the same problem in the complementary case ρ &lt; 0 {\rho&lt;0} .

Long time decay analysis of complex-valued fractional abstract evolution equations with delay

The asymptotic stability and long time decay rates of solutions to linear Caputo time-fractional ordinary differential equations are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of time-fractional ordinary differential equations decay only polynomially, leading to the so-called Mittag-Leffler stability, which were already extended to linear fractional delay differential equations (FDDEs) with real constants by Čermák et al. (2017) [17]. This paper is devoted to the asymptotical stability and long time decay rates of the solutions for a class of fractional delay abstract evolution equations with complex-valued coefficients. By applying the spectral decomposition method, we reformulate the fractional delay abstract evolution equations to an infinite-dimensional FDDEs. Through the root locus technique, we develop a necessary and sufficient stability condition in a coefficient criterion for FDDEs. At the same time, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long time decay rates of the solutions for both linear FDDEs and fractional delay abstract evolution equations with complex-valued coefficients are O(t−α). Finally, several application models including both time and time-space fractional Schrödinger equations are presented to illustrate the effectiveness of our proposed criteria.

On the topological size of the class of Leray solutions with algebraic decay

AbstractIn 1987, Michael Wiegner in his seminal paper (J. Lond. Math. Soc. (2), 35 (1987) 303–313) provided an important result regarding the energy decay of Leray solutions to the incompressible Navier–Stokes in : if the associated Stokes flows had their norms bounded by for some , then the same would be true of . The converse also holds, as shown by Skalák (J. Math. Fluid Mech. 16 (2014) 431–446) and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier–Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two‐sided bounds of the form .

Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms

Viscoelastic damping phenomena are ubiquitous in diverse kinds of wave motions of nonlinear media. This arouses extensive interest in studying the existence, the finite time blow-up phenomenon and various large time behaviors of solutions to viscoelastic wave equations. In this paper, we are concerned with a class of variable coefficient coupled quasi-linear wave equations damped by viscoelasticity with a long-term memory fading at very general rates and possibly damped by friction but provoked by nonlinear interactions. We prove a local existence result for solutions to our concerned coupled model equations by applying the celebrated Faedo-Galerkin scheme. Based on the newly obtained local existence result, we prove that solutions would exist globally in time whenever their initial data satisfy certain conditions. In the end, we provide a criterion to guarantee that some of the global-in-time-existing solutions achieve energy decay at general rates uniquely determined by the fading rates of the memory. Compared with the existing results in the literature, our concerned model coupled wave equations are more general, and therefore our theoretical results have wider applicability. Modified energy functionals (can also be viewed as certain Lyapunov functionals) play key roles in proving our claimed general energy decay result in this paper.

Decay of solutions of the wave equation in cosmological spacetimes—a numerical analysis

We numerically evolve spherically symmetric solutions to the linear wave equation on some expanding Friedmann–Lemaître–Robertson–Walker spacetimes and study the respective asymptotics for large times. We find a quantitative relation between the expansion rate of the underlying background Universe and the decay rate of linear waves, also in the context of spatially-hyperbolic spacetimes, for which rigorous proofs of decay rates are not generally known. A prominent role in the decay mechanism is shown to be played by tails, i.e. scattered waves propagating in the interior of the lightcone.

Study of CO2 absorption by low salinity aqueous solutions of specific salts in the absence and presence of carbonate rocks

Fossil-fuels combustion is currently the main driving force of industrialization; however, it has resulted in CO2 emission that challenges sustained socioeconomic growth. One way to reduce CO2 emissions is to sequester it into saline aquifers or depleted hydrocarbon reservoirs. In this study, CO2 absorption in different aqueous salt solutions is elucidated. CO2 absorption in different salines is also simulated using PHREEQC software to predict the concentration of different species formed by saline-CO2 interactions at equilibrium. Investigated parameters include concentration of different salts at low salinity condition, type of cations and anions, temperature, and carbonate rock on the CO2 absorption and storage in aqueous media at high pressures. Results revealed that both initial absorption rate and final absorbed amount of CO2 in aqueous salt solution decay in the order of MgSO4 > Na2SO4 > CaCl2 > MgCl2 > NaCl. For a given anion in the dissolved salt, the presence of divalent cations promotes CO2 absorption more than that of monovalent cations. Also, for equivalence cations the one with a higher charge density promotes CO2 absorption. It is also evident that for a given cation in the dissolved salt the associated anion which has a higher charge density hinders CO2 absorption more than that with a lower charge density. Similar pattern is maintained for CO2 absorption in saturated carbonate rock. PHREEQC simulations revealed that concentrations of precipitated carbonate salts in CO2-saturated Na2SO4 and NaCl brines are higher than those of the other brines. Moreover, increasing temperature increases concentration of precipitated carbonate salts in CO2-saturated MgSO4, CaCl2, and MgCl2 brines while it has an opposite effect on precipitated carbonate salts formed due to CO2 absorption in Na2SO4 and NaCl brines.

Ground state solutions for Choquard equations with Bessel potential

Firstly, we study the existence, regularity and decay of positive solutions for the elliptic system { − Δ u + u = vg ( u ) in R N , − Δv + v = G ( u ) in R N , where N ≥ 3 , g satisfies Berestycki-Lions type conditions. In Sobolev space, the system can be transformed into the Choquard equations with Bessel potential − Δ u + u = ( g 2 ⋆ G ( u ) ) g ( u ) in R N , where g 2 is the fundamental solution of the linear operator − Δ + I on R N . Moreover, by assuming that g is nondecreasing, we obtain the radial symmetry and monotonic property of positive solutions by moving plane method, and the existence of ground state solutions by non-Nehari manifold method.

Global existence, energy decay, and blowup of solutions for a wave equation type p-Laplacian with memory term and dynamic boundary conditions

Global existence, energy decay, and blowup of solutions for a wave equation type p-Laplacian with memory term and dynamic boundary conditions

Two dimensional wave-Klein–Gordon equations with a below-critical nonlinearity

In this paper we investigate the small data global existence and pointwise decay of solutions to two systems of coupled wave-Klein–Gordon equations in two spatial dimensions. In particular, we consider critical (in the sense of time decay) semilinear nonlinearities for the wave equation and below-critical semilinear nonlinearities for the Klein–Gordon equation, a situation that has not been studied before in the context of coupled wave and Klein–Gordon equations. An interesting feature of our two systems is that the below-critical nonlinearity causes the Klein–Gordon field to lose its linear behaviour close to the light cone, even though it enjoys optimal time decay.

On the speed of decay of solutions to some partial differential equations

In this paper we show that it is possible to estimate the speed of decay (as t→+∞) of a function u(t) simply by proving that it satisfies certain integral inequalities. Moreover, we study the relationship between these integral inequalities and the phenomenon of the extinction in finite time proving that the extinction appears for a suitable choice of the exponents in these inequalities. Finally, we give same easy example of possible applications of these general results to the description of the asymptotic behavior of the solutions to some different evolution problems.