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In this work, we introduce and study two problems for diffusion equations with the distributed order fractional derivatives including the initial value problem and the terminal value problem. For the initial value problem, we establish some existence results and Hölder regularity for the mild solution. On the other hand, we also show the existence results and a decay estimate of the mild solution for the terminal value problems. Especially, the polynomial decay of the solutions to the terminal value problems is firstly included when the source function is equal to zero. [ABSTRACT FROM AUTHOR]

This paper deals with the coupled fourth-order parabolic equations, subject to Dirichlet or Navier boundary conditions. We combine the generalized potential well method, the concave method, with the auxiliary methods to classify the initial energy about the existence of blow-up solutions and global solutions. Moreover, the decay estimate of global solutions and the bounds of blow-up time and rate are determined, respectively. [ABSTRACT FROM AUTHOR]

We investigate the dissipative structure for the system of Moore–Gibson–Thompson (MGT) thermoelasticity in the whole space. To analyze the dissipative structure, it is very useful to rewrite the equations into a symmetric hyperbolic system and apply the so‐called stability condition. When we rewrite our system into the symmetric hyperbolic form in the multidimensional case, it is important to take the constraint conditions into account. Indeed, the stability condition with the constraint conditions guarantees the dissipative property for our system in some cases. In this paper, we introduce the stability condition with constraints for the general problem and apply this argument to the system of MGT thermoelasticity. Furthermore, we discuss the optimality of the decay estimates we obtain, together with the no regularity‐loss phenomenon. [ABSTRACT FROM AUTHOR]

We study the global well-posedness of the simplified Ericksen-Leslie system with zero viscosity on the periodic domain \mathbb {T}^2. Precisely, we prove the global existence and uniqueness of smooth solution to Ericksen-Leslie system if the initial data (u_0,\nabla d_0) is small in H^4(\mathbb {T}^2)\times H^4(\mathbb {T}^2). Furthermore, we derive the time decay estimate of \nabla d in H^1(\mathbb {T}^2). [ABSTRACT FROM AUTHOR]

In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing nonlinear Schrödinger equation with various (deterministic and random) initial data. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see [C. Fan and Z. Zhao, Discrete Contin. Dyn. Syst., 41 (2021), pp. 3973--3984] and the references therein), and, moreover, we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the L¹ -data assumption (see [C. Fan and Z. Zhao, Proc. Amer. Math. Soc., 151 (2023), pp. 2527--2542] for the necessity of the L¹-data assumption). [ABSTRACT FROM AUTHOR]

In this paper we deal with the problem of strong stabilisation for a class of distributed semilinear systems by means of a nonlinear feedback control. The systems under consideration evolve in a Hilbert space and present periodic time-varying state delay. We first give a proof for the existence and uniqueness of the solution for the considered systems. Then, under a null controllability condition, we establish the stabilisation result and provide an explicit optimal decay rate estimate. The particular case of such systems for which the linear part generates a differentiable semigroup is also investigated. Some illustrating applications to hyperbolic and parabolic equations are displayed. Finally, a conclusion and some perspectives are given. [ABSTRACT FROM AUTHOR]

We aim at the stability of time-dependent motions, such as time-periodic ones, of a rigid body in a viscous fluid filling the exterior to it in 3D. The fluid motion obeys the incompressible Navier–Stokes system, whereas the motion of the body is governed by the balance for linear and angular momentum. Both motions are affected by each other at the boundary. Assuming that the rigid body is a ball, we adopt a monolithic approach to deduce L q – L r decay estimates of solutions to a non-autonomous linearized system. We then apply those estimates to the full nonlinear initial value problem to find temporal decay properties of the disturbance. Although the shape of the body is not allowed to be arbitrary, the present contribution is the first attempt at analysis of the large time behavior of solutions around nontrivial basic states, that can be time-dependent, for the fluid–structure interaction problem and provides us with a stability theorem which is indeed new even for steady motions under the self-propelling condition or with wake structure. [ABSTRACT FROM AUTHOR]

In this paper, we aim to study the feedback stabilization of an infinite-dimensional semilinear system evolving in reflexive Banach state space. The concept of bounded control is also investigated in the realistic domain. Sufficient conditions for appropriate feedback control to ensure strong and weak stabilization are given. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the problem of feedback stabilization for distributed bilinear systems with discrete delay evolving on a Hilbert state space. Firstly, we prove the existence and uniqueness of the mild solution of system at hand. Secondly, we study the weak and strong stabilization. More precisely, we provide some sufficient conditions to ensure the weak and strong stabilization for the delayed bilinear system. Moreover, in the case of the strong stabilization, an explicit decay estimate is established. Applications to wave and heat equations with delayed bilinear parts are considered. [ABSTRACT FROM AUTHOR]

We are concerned with the global well-posedness of rate type viscoelastic fluids system (1.1)–(1.2). First, we prove the global existence and decay estimate of solution to (1.1)–(1.2) with stress diffusion near a constant state in H N (R 3) (N ≥ 4). We only assume that the H 3 -norm of initial data is small, but the L 2 -norm of the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev or Besov spaces, we obtain the optimal decay rates of the solution and its higher order derivatives. Next, we establish the local existence of smooth solution to (1.1)–(1.2) without stress diffusion and present a blow-up criterion in R 2. From which and the Bony decomposition, we prove the global existence of smooth solution. [ABSTRACT FROM AUTHOR]

We study the stability for a coupled system of second-order evolution equations with indirect memory-damping, which arises from the theory of viscoelasticity and takes the Timoshenko systems as concrete examples. What concerns us is how to efficiently control the memory kernels to achieve the desired stabilities of the whole system. With the help of the memory and coupling effects in the system, we establish some optimal stability results for the energy of the system under weak conditions. Especially, we do not need the usual nonnegative/nonincreasing restrictions on the memory kernels in our results. For the Timoshenko systems, we obtain the optimal decay rate under minimal assumptions on memory kernels and the best (so far) exponential stability result in the case of equal wave speeds. As can be seen, to overcome the difficulties in obtaining these results, novel ideas and techniques are presented and utilized in our proofs. [ABSTRACT FROM AUTHOR]

In this article, we investigate the initial-boundary value problem for a class of finitely degenerate semilinear parabolic equations with singular potential term. By applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution. On the other hand, by constructing a family of potential wells, we prove the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.

The heat semigroup { T (t) } t ≥ 0 defined on homogeneous Besov spaces B ˙ p , q s (R n) is considered. We show the decay estimate of T (t) f ∈ B ˙ p , 1 s + σ (R n) for f ∈ B ˙ p , ∞ s (R n) with an explicit bound depending only on the regularity index σ > 0 and space dimension n. It may be regarded as a refined result compared with that of the second author (Takeuchi in Partial Differ Equ Appl Math 4:100174, 2021). As a result of the refined decay estimate, we also improve a lower bound estimate of the radius of convergence of the Taylor expansion of T (·) f in space and time. To refine the previous results, we show explicit pointwise estimates of higher order derivatives of the power function | ξ | σ for σ ∈ R . In addition, we also refine the L 1 -estimate of the derivatives of the heat kernel. [ABSTRACT FROM AUTHOR]

In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schrödinger equations assuming data in L 1 ∩ H 3. 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. [ABSTRACT FROM AUTHOR]

In this paper, the initial-boundary value problems for the time-fractional degenerate evolution equations are considered. Firstly, in the linear case, we obtain the optimal rates of decay estimates of the solutions. The decay estimates are also established for the time-fractional evolution equations with nonlinear operators such as: p-Laplacian, the porous medium operator, degenerate operator, mean curvature operator and Kirchhoff operator. At the end, some applications of the obtained results are given to derive the decay estimates of global solutions for the time-fractional Fisher-KPP-type equation and the time-fractional porous medium equation with the nonlinear source. [ABSTRACT FROM AUTHOR]

In this article, we study the following Klein-Gordon-Maxwell system: −Δu−(2ω+ϕ)ϕu=g(u),inR3,Δϕ=(ω+ϕ)u2,inR3,\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}-\Delta u-\left(2\omega +\phi )\phi u=g\left(u),\hspace{1.0em}{\rm{in}}\hspace{1em}{{\mathbb{R}}}^{3},\hspace{1.0em}\\ \Delta \phi =\left(\omega +\phi ){u}^{2},\hspace{1.0em}{\rm{in}}\hspace{1em}{{\mathbb{R}}}^{3},\hspace{1.0em}\end{array}\right. where ω\omega is a constant that stands for the phase; uu and ϕ\phi are unknowns and gg satisfies the Berestycki-Lions condition [Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313–345; Nonlinear scalar field equations. II. Existence of infinitelymany solutions, Arch. Rational Mech. Anal. 82 (1983), 347–375]. The Klein-Gordon-Maxwell system is a model describing solitary waves for the nonlinear Klein-Gordon equation interacting with an electromagnetic field. By using variational methods and some analysis techniques, the existence of positive solution and multiple solutions can be obtained. Moreover, we study the properties of decay estimates and asymptotic behavior for the positive solution.

In this paper, we study a fourth order parabolic equation modeling epitaxial thin film growth. By using the potential well method and some inequality techniques, we obtain the decay estimate of weak solutions. Meanwhile, the blow-up time is estimated from above and below. The blow-up rate is also derived.

We show the upper and lower bounds of convergence rates for strong solutions of the 3D non-Newtonian flows associated with Maxwell equations under a large initial perturbation. [ABSTRACT FROM AUTHOR]

This paper focuses on the stability problem and decay estimates for two classes of three-dimensional (3D) magneto-micropolar equations with mixed partial viscosity. When μ 3 = ν 3 = γ 1 = γ 2 = 0 and χ Δ u replaced by χ u in (1.2), by fully exploiting the structure of the system, and the method of bootstrapping argument, we prove the global stability of solutions to this system with initial data small in H 2 (R 3) . Furthermore, for these global solutions with initial data in H s (R 3) (s ≥ 3) being large, we obtain their global H s (R 3) (s ≥ 3) bound which is independent of time. In addition, we obtain the global existence of solutions for small initial data and the decay estimates of these solutions to 3D magneto-micropolar equations with mixed partial viscosity [namely μ 3 = γ 1 = γ 2 = 0 and χ Δ u replaced by χ u in (1.2)]. [ABSTRACT FROM AUTHOR]

In this note, we show the existence of a special solution u to defocusing cubic NLS in 3d, which lives in H^{s} for all s>0, but scatters to a linear solution in a very slow way. We prove for this u, for all \epsilon >0, one has \sup _{t>0}t^{\epsilon }\|u(t)-e^{it\Delta }u^{+}\|_{\dot {H}^{1/2}}=\infty. Note that such a slow asymptotic convergence is impossible if one further pose the initial data of u(0) be in L^{1}. We expect that similar construction hold the for other NLS models. It can been seen the slow convergence is caused by the fact that there are delayed backward scattering profile in the initial data, we also illustrate why L^{1} condition of initial data will get rid of this phenomena. [ABSTRACT FROM AUTHOR]

We study an optimal L 2 -decay of solutions to the Cauchy problem of the n -dimensional nonlinear Schrödinger equations with a long-range dissipative nonlinearity. Our purpose is to show that the optimal L 2 -decay of dissipative solutions is given by (log ⁡ t) − n / 2 in higher space dimensions. [ABSTRACT FROM AUTHOR]

This paper discusses the Cauchy problem for a semilinear homogeneous, isotropic Mindlin–Timoshenko plate in R 2 with irrotational filament angle field. Applying Kawashima’s energy method in the frequency space, the decay rate for the semigroup associated with the linearized system is established. Moreover, the existence and uniqueness of local solution are proved with Banach’s fixed point theorem. Finally, the global existence and uniqueness of classical solution along with polynomial decay are proved via the barrier method. [ABSTRACT FROM AUTHOR]

In this paper, we study a fourth order parabolic equation modeling epitaxial thin film growth. By using the potential well method and some inequality techniques, we obtain the decay estimate of weak solutions. Meanwhile, the blow-up time is estimated from above and below. The blow-up rate is also derived. [ABSTRACT FROM AUTHOR]

Abstract In this article, we consider a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity. We prove explicit and general decay rate results of the solution to the viscoelastic plate equation with past history. Convex properties, logarithmic inequalities, and generalized Young’s inequality are mainly used to prove the decay estimate.

In this paper, we investigate the Cauchy problem related to the standard linear solid model with thermoelasticity of type III in the whole space. By using semigroup theory, we first establish the well-posedness result under certain assumptions on the parameters. By applying the energy method in the Fourier space, we then prove the optimal decay estimate results for both the non-critical and critical cases. Specifically, the decay property of the system is not of the regularity-loss type. In addition, we study the asymptotic expansion of the eigenvalues to analyze the optimality of the decay results. [ABSTRACT FROM AUTHOR]

We consider the Keller-Segel-Navier-Stokes system which is considered in bounded domain Ω ⊂ ℝN (N ∈ {2, 3}) with smooth boundary, where with δ ∈ (0, 1). We show that if the initial data , , and is small enough, an associated initial-boundary value problem possesses a global classical solution which decays to the constant state () exponentially with . [ABSTRACT FROM AUTHOR]

We generalize Holley–Stroock's perturbation argument from commutative to finite dimensional quantum Markov semigroups. As a consequence, results on (complete) modified logarithmic Sobolev inequalities and logarithmic Sobolev inequalities for self-adjoint quantum Markov processes can be used to prove estimates on the exponential convergence in relative entropy of quantum Markov systems which preserve a fixed state. This leads to estimates for the decay to equilibrium for coupled systems and to estimates for mixed state preparation times using Lindblad operators. Our techniques also apply to discrete time settings, where we show that the strong data processing inequality constant of a quantum channel can be controlled by that of a corresponding unital channel. [ABSTRACT FROM AUTHOR]

This work deals with a class of nonlinear viscoelastic wave equations with strong damping and variable exponent sources. The main interest of this paper compared to many previous works in the literature is able to use the idea of potential well method to study both the finite time blow-up for solutions starting from the unstable sets and decay estimate for global solutions starting in the potential wells. [ABSTRACT FROM AUTHOR]

The aim of this paper is to utilize special skills and potential well method to deal with questions of local and global well-posedness to a nonlinear diffusion equation driven by the p-Laplace operator, which reveals a subtle relationship between singular potential and nonlinear operator, and further generalizes some previous early results about local existence of the weak solution from a class of evolution equations involved the linear dissipative term and singular potential. Specifically, the results of stability and blow-up of solutions are established for the first time to this initial boundary value problem with singular potential. [ABSTRACT FROM AUTHOR]

In this paper, the $L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness of the bilinear oscillatory integral along parabola \begin{equation*} T_\beta(f, g)(x)=p.v.\int_{{\mathbb R}} f(x-t)g(x-t^{2})\,{\rm e}^{i |t|^{\beta}}\,\frac{{\rm d}t}{t} \end{equation*} is set up, where β  > 1 or β  < 0, $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $\frac{1}{2}\lt r\lt\infty$ , p  > 1 and q  > 1. The result for the case β  < 0 extends the $L^\infty\times L^2\to L^2$ boundedness obtained by Fan and Li (D. Fan and X. Li, A bilinear oscillatory integral along parabolas, Positivity 13(2) (2009), 339–366) by confirming an open question raised in it. [ABSTRACT FROM AUTHOR]

We are concerned with a system of equations governing the evolution of isothermal, viscous, and compressible fluids of the Korteweg type, which is used to describe a two-phase liquid–vapor mixture. It is found that there is a "regularity-gain" dissipative structure of linearized systems in case of zero sound speed P ′ (ρ *) = 0 , in comparison with the classical compressible Navier–Stokes equations. First, we establish the global-in-time existence of strong solutions in hybrid Besov spaces by using Banach's fixed point theorem. Furthermore, we prove that the global solutions with critical regularity are Gevrey analytic in fact. Secondly, based on Gevrey's estimates, we obtain uniform bounds on the growth of the analyticity radius of solutions in negative Besov spaces, which lead to the optimal time-decay estimates of solutions and their derivatives of arbitrary order. [ABSTRACT FROM AUTHOR]

In this article, we consider a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity. We prove explicit and general decay rate results of the solution to the viscoelastic plate equation with past history. Convex properties, logarithmic inequalities, and generalized Young's inequality are mainly used to prove the decay estimate. [ABSTRACT FROM AUTHOR]

In this paper, we study the one-dimensional models of blood flow arising from the hemodynamics of aorta, which are derived from the averaging of the Navier–Stokes equations. We establish the global well-posedness and long-time behavior of the viscid 1D models of blood flow in the Sobolev space framework, where a general parabolic velocity profile is considered. Precisely speaking, we prove the global existence of smooth solution when the initial data is sufficiently small. Moreover, by combining the time-weighted energy estimates with the Green function method, we obtain the optimal time decay rate in L p (2 ≤ p ≤ ∞) norm. In addition, one can see the area-averaged axial velocity decays faster than the cross-sectional area of vessel from the Green function of the linearized system. This observation is essential to study the decay rates of our nonlinear system. [ABSTRACT FROM AUTHOR]

In this paper, we study global existence and blow-up of solutions to the viscous Moore-Gibson-Thompson (MGT) equation with the nonlinearity of derivative-type |ut|p. We demonstrate global existence of small data solutions if p>1+4/n (n≤6) or p≥2−2/n (n≥7), and blow-up of nontrivial weak solutions if 1

We are concerned with extinction properties for a p-Laplacian equation with gradient source and absorption terms under nonlinear boundary condition. Based on the integral norm estimate method and the technique of differential inequalities, we establish some extinction results and the precise decay estimates of extinction solutions for this problem. Furthermore, we believe that the arguments used in this paper can also be applicable to the problem with nonlocal source instead of a gradient source. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider the Cauchy problem for the dissipative nonlinear Schrödinger equation with a critical cubic nonlinearity in one space dimension. We show the global uniform bound of dissipative solutions in the Gevrey class and its L 2 -decay order without any restriction of the size of smooth initial data. [ABSTRACT FROM AUTHOR]

The purpose of this work is to study the strong stabilization of the state fractional spatial derivative of order q ∈ (0 , 1) , for a class of semi-linear dynamical systems. First of all, we establish the existence and the uniqueness of the global solution for such a class of systems. Also, we develop sufficient conditions for the strong stabilization with decay estimate of the state fractional spatial derivative. Moreover, under weaker conditions, we determine an alternative feedback control that ensures the strong stabilization of the considered output. Furthermore, we investigate the fractional output stabilization by considering the minimization of a suitably chosen functional cost. Finally, we present a numerical example to illustrate the applicability of our obtained results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop the general L p theory for the barotropic compressible Navier-Stokes equations with the free boundary condition in the exterior domain in R N (N ≥ 3). By the spectral analysis, we obtain the classical L p - L q decay estimate for the linearized model problem (with variable coefficients) in view of the partial Lagrangian transformation. [ABSTRACT FROM AUTHOR]

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D-nonlinear Schrödinger equations with a sub-critical dissipative power nonlinearity. Our aim is to obtain two results. One asserts that, if a global solution decays more rapidly than t - d with d = 1 / (p - 1) - 1 / 2 in L 2 , then it must be a trivial solution. The other one shows the existence of a solution decaying just at the rate of t - d in L 2 . [ABSTRACT FROM AUTHOR]

This paper studies the stability and decay estimates of solutions to the two-dimensional (2D) magneto-micropolar fluid equations with partial dissipation. We first establish the L 2 -decay estimates for global solutions and their derivative with initial data in L 1 (R 2) . Furthermore, we show the global stability of these solutions in H s (R 2) , and the decay rates of these global solutions and their higher derivatives when the initial data belongs to the negative Sobolev space H ˙ - l (R 2) (for each 0 ≤ l < 1 ). [ABSTRACT FROM AUTHOR]

In this paper, we consider a wave equation with memory term and time‐varying delayed velocity term. By some subtle analysis techniques, assuming memory kernel function with a minimal and general condition, we establish a general uniform energy decay result for the system. The condition here substantially weakens the conditions in earlier literature. From the result, we can give the explicit energy decay rates directly if the memory kernel is expressed explicitly. The result is new even for the case that memory kernel decays polynomially. [ABSTRACT FROM AUTHOR]

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than (log t) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of (log t) − 1 / 2 in L 2 . [ABSTRACT FROM AUTHOR]

In this paper, we discuss global existence, boundness, blow-up and extinction properties of solutions for the Dirichlet boundary value problem of the $ p $-Laplacian equations with logarithmic nonlinearity $ u_{t}-{\rm{div}}(|\nabla u|^{p-2}\nabla u)+\beta|u|^{q-2}u = \lambda |u|^{r-2}u\ln{|u|} $, where $ 1 < p < 2 $, $ 1 < q\leq2 $, $ r > 1 $, $ \beta, \lambda > 0 $. Under some appropriate conditions, we obtain the global existence of solutions by means of the Galerkin approximations, then we prove that weak solution is globally bounded and blows up at positive infinity by virtue of potential well theory and the Nehari manifold. Moreover, we obtain the decay estimate and the extinction of solutions.