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In this paper, we are concerned with the well-posedness and large time behavior of Cauchy problem for Cahn–Hilliard equation in R n ( n ∈ Z + , n ≥ 3 ). First, based on the higher-order norm estimates of solutions and the mollifier technique, we obtain the local well-posedness of strong solutions. Then, by using pure energy method, standard continuity argument together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates provided that the H n 2 + 1 -norm of initial data is sufficiently small. [ABSTRACT FROM AUTHOR]

Using weighted spaces, we establish a general decay rate properties of solutions as T w for a coupled system of viscoelastic wave equations in Rn under some conditions on g1, g2, We exploit a density function to introduce weighted spaces for solutions and using an appropriate Lyapunov function. [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]

The decay rate of solutions to a Bresse system in thermoelasticity of type III with respect to the distributed delay term is the subject of this study. We demonstrate our major finding utilising the energy approach in the Fourier space. [ABSTRACT FROM AUTHOR]

In this work, we deal with a one-dimensional Cauchy problem in Timoshenko system with thermal effect and damping term. The heat conduction is given by the theory of Lord-Shulman. We prove that the dissipation induced by the coupling of the Timoshenko system with the heat conduction of Lord-Shulman's theory alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow. [ABSTRACT FROM AUTHOR]

We consider equations describing the thermoelastic behavior of plates modeled in the Green-Lindsay sense. This is done with two different type of couplings of the fourth-order plate Kirchhoff-type plate equation to a second-order heat equation of Cattaneo type, once of second, and once of first order. We investigate both systems for bounded domains and for the Cauchy problem, asking for exponential stability in bounded domains resp. polynomial decay rates for the Cauchy problem. It turns out that one system is exponentially stable, while the other is not, and that, in correspondence, one does not have and the other one has regularity loss in the Cauchy problem. This provides a new interesting example where the different couplings lead to qualitatively different behavior, as known before for classical thermoelastic plates, for Timoshenko systems, for porous elasticity or for plates with two temperatures, with Fourier resp. Cattaneo heat conduction. The optimality of the decay rates obtained is also proved. [ABSTRACT FROM AUTHOR]

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We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space ℝN. We show that given a weighted Lp-space $L_w^p({\mathbb {R}}^N)$ with 1 ⩽ p < ∞ and a fast growing weight w, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_w^p({\mathbb {R}}^N)$ with the following property: given an arbitrary positive integer m there exists nm > 0 such that, if the initial data f belongs to the closed linear span of en with n ⩾ nm, then the decay rate of the solution of the problem is at least t−m for large times t. The result generalizes the recent study of the authors concerning the classical linear heat equation. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited m. [ABSTRACT FROM AUTHOR]

In this paper, we show that bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the form u t + div f (x , t , u) = div (| u | α ∇ u) , x ∈ R n , t > 0 , where α > 0 is constant, decrease to zero, under fairly broad conditions on the advection flux f . Besides that, we derive a time decay rate for these solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Moore–Gibson–Thompson equation in R N , which is a third order in time equation that arises in viscous thermally relaxing fluids and also in viscoelastic materials (then under the name of standard linear viscoelastic model). First, we use some Lyapunov functionals in the Fourier space to show that, under certain assumptions on some parameters in the equation, a norm related to the solution decays with a rate (1 + t) - N / 4 . Since the decay of the previous norm does not give the decay rate of the solution itself then, in the second part of the paper, we show an explicit representation of the solution in the frequency domain by analyzing the eigenvalues of the Fourier image of the solution and writing the solution accordingly. We use this eigenvalues expansion method to give the decay rate of the solution (and also of its derivatives), which results in (1 + t) 1 - N / 4 for N = 1 , 2 and (1 + t) 1 / 2 - N / 4 when N ≥ 3 . [ABSTRACT FROM AUTHOR]

This paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in ℝN (N ∈ ℕ+), provided that the initial datum is sufficiently small. In addition, the Lp-norm ( 1 ≤ p ≤ ∞) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in ℝN (N ∈ ℕ+), provided that the initial datum is sufficiently small. In addition, the Lp-norm ( 1 ≤ p ≤ ∞) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied. [ABSTRACT FROM AUTHOR]

In this article, we study the energy decay for the thermoelastic Bresse system in the whole line with two dissipative mechanisms, given by heat conduction (Types I and III). We prove that the decay rate of the solutions are very slow. More precisely, we show that the solutions decay with the rate of (1+t)-1/8 in the L²-norm, whenever the initial data belongs to L¹(R)∩Hs(R) for a suitable s. The wave speeds of propagation have in uence on the decay rate with respect to the regularity of the initial data. This phenomenon is known as regularity-loss. The main tool used to prove our results is the energy method in the Fourier space. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Bresse system coupled with the Green and Naghdi type-III theory of heat conduction. We prove that the decay rate of the solution is very slow. In fact, we show that the-norm of the solution decays with the rate ofsimilar to the one obtained for the Timoshenko system in thermoelasticity of type III. Furthermore, we found that the wave speeds of the equations control the decay rate of the solution with respect to the regularity of the initial data. In addition, we prove that the solution enjoys the property of regularity loss. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stability and decay rate of viscous contact wave for the Cauchy problem of one-dimensional compressible Navier–Stokes equations with radiation under the zero mass condition. we consider the case that the constitutive relations for the pressure p , the specific internal energy e , the specific volume v and the absolute temperature θ are given by p = R θ / v + a θ 4 / 3 , e = C v θ + a v θ 4 , with R > 0 , C v > 0 , and a > 0 being the perfect gas constant, the specific heat and the radiation constant, respectively. Our main idea is to use the smallness of the strength of the contact wave to control the possible growth of its solutions induced by the nonlinearity of the system and interactions between the solutions and the contact wave. The key point in our analysis is to obtain the uniform convergence rate of a contact wave pattern (1 + t) − 5 8 ln 1 2 (2 + t) by using a new inequality (see Lemma 2.3) and a weighted characteristic energy estimate. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the solutions of the one-dimensional Cauchy problem in Timoshenko system with thermal effect. The heat conduction is given by the type III theory of Green and Naghdi. We prove that the dissipation induced by the heat conduction alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow and, in the case of nonequal wave speeds, are of regularity-loss type. This paper solves the open problem stated in [10] and shows that the stability of the solution holds without any additional mechanical damping term. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of the attraction–repulsion Keller–Segel chemotaxis model { u t = Δ u − ∇ ⋅ ( χ u ∇ v ) + ∇ ⋅ ( ξ u ∇ w ) , x ∈ R n , t > 0 , v t = Δ v + α u − β v , x ∈ R n , t > 0 , w t = Δ w + γ u − δ w , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , x ∈ R n . Here all parameters χ , ξ , α , β , γ and δ are positive. When repulsion cancels attraction (i.e., ξ γ = χ α ), the existence of global classical solution is established with large initial data in two or three spatial dimensions. Moreover, we show that the classical solution decays to zero as t → ∞ and behaves like the heat kernel. [ABSTRACT FROM AUTHOR]

In this paper, we first prove the global existence for the Cauchy problem of the Navier–Stokes–Nernst–Planck–Poisson (NSNPP) system in dimension two. Then we establish the L 2 decay estimates of the solutions by using the Fourier splitting method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the Cauchy problem for a system of elastic solids with voids. First, we show that a linear porous dissipation leads to decay rates of regularity-loss type of the solution. We show some decay estimates for initial data in . Furthermore, we prove that by restricting the initial data to be in and , we can derive faster decay estimates of the solution. Second, we show that by adding a viscoelastic damping term, then we gain the regularity of the solution and obtain the optimal decay rate. [Copyright &y& Elsevier]

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]

In this paper, we investigate the decay property of a Timoshenko system in thermoelasticity of type III in the whole space where the heat conduction is given by the Green and Naghdi theory. Surprisingly, we show that the coupling of the Timoshenko system with the heat conduction of Green and Naghdi's theory slows down the decay of the solution. In fact we show that the L²-norm of the solution decays like (1 + t)- 1/8, while in the case of the coupling of the Timoshenko system with the Fourier or Cattaneo heat conduction, the decay rate is of the form (1 + t)- 1/4 [25]. We point out that the decay rate of (1 + t)- 1/8 has been obtained provided that the initial data are in L¹ (ℝ) ∩ Hs(ℝ), (s ≥ 2). If the wave speeds of the first two equations are different, then the decay rate of the solution is of regularity-loss type, that is in this case the previous decay rate can be obtained only under an additional regularity assumption on the initial data. In addition, by restricting the initial data to be in Hs (ℝ) ∩ L1,γ (ℝ) with γ ∈ [0, 1], we can derive faster decay estimates with the decay rate improvement by a factor of t-γ/4. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of two models of the theory of heat conduction with three-phase-lag. Under appropriate assumptions on the material parameters, we show the optimal decay rate of the L²-norm of solutions. More precisely, we prove that in each model the L²-norm of the solution is decaying with the rate (1 + t)-1/4 for initial data in L¹(ℝ). This decay rate is similar to the one of the heat kernel. Some faster decay rates have been also given for some weighted initial data in L¹(ℝ). [ABSTRACT FROM AUTHOR]

Abstract: We consider the Cauchy problem for the damped wave equation with space–time dependent potential and absorbing semilinear term . Here, with , and . Using the weighted energy method, we can obtain the decay rate of the solution, which is almost optimal in the case . Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, -estimates of solutions for damped wave equations with space–time dependent damping term, J. Differential Equations 248 (2010) 403–422], we believe that is a critical exponent. Note that when , coincides to the Fujita exponent . The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects with initial data where α and ν are positive constants such that , . Under the assumption that is sufficiently small, we show that if the initial data is a small perturbation of the parabolic system defined by (2.4) which are obtained by the convection–diffusion equations (2.1), and solutions to Cauchy problem (E) and (I) tend asymptotically to the convection–diffusion system with exponential rates. Precisely speaking, we derive the asymptotic profile of (E) by Gauss kernel as follows: The same problem was studied by Tang and Zhao [S.Q. Tang, H.J. Zhao, Nonlinear stability for dissipative nonlinear evolution equations with ellipticity, J. Math. Anal. Appl. 233 (1999) 336–358], Nishihara [K. Nishihara, Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity, Z. Angew. Math. Phys. 57 (4) (2006) 604–614] for the case of . [Copyright &y& Elsevier]

In this paper, we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects: with initial data where α and ν are positive constants such that α < 1, ν < 4 α(1 − α). Under the assumption that $$ \left| \psi_{ + } - \psi_{ - } \right| + \left| \theta _{ + } - \theta _{ - } \right| $$ is sufficiently small, we show the global existence of the solutions to Cauchy problem (E) and (I) if the initial data is a small perturbation. And the decay rates of the solutions with exponential rates also are obtained. The analysis is based on the energy method. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Cauchy problem for the damped wave equation with absorption The behavior of u as is expected to be the same as that for the corresponding heat equation which has the similarity solution with the form depending on provided that p is less than the Fujita exponent . In this paper, as a first step, if and the data decays exponentially as without smallness condition, the solution is shown to decay with rates as , those of which seem to be reasonable, because the similarity solution has the same decay rates as (∗). For the proof, the weighted -energy method will be employed with suitable weight, similar to that in Todorova and Yordanov [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464–489]. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the global existence and asymptotic behaviors of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects: with initial data where α and ν are positive constants such that , . Under the assumption that is sufficiently small, we show that if the initial data is a small perturbation of the diffusion waves defined by (2.5) which are obtained by the diffusion equations (2.1), solutions to Cauchy problem (E) and (I) tend asymptotically to those diffusion waves with exponential rates. The analysis is based on the energy method. The same problem was studied by Tang and Zhao [J. Math. Anal. Appl. 233 (1999) 336–358] for the case of . [Copyright &y& Elsevier]