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We consider the abstract MGT-Fourier system $${\left\{ \begin{array}{ll} u_{ttt}+\alpha u_{tt}+\beta A^\varrho u_t+\gamma A^\varrho u = \eta A \theta \\ \theta _t+\kappa A \theta = -\eta A u_{tt} - \eta \alpha A u_t \end{array}\right. }$$ u ttt + α u tt + β A ϱ u t + γ A ϱ u = η A θ θ t + κ A θ = - η A u tt - η α A u t in the subcritical regime $$\beta >\gamma /\alpha$$ β > γ / α , where the operator A is strictly positive selfadjoint. For any fixed $$\varrho \in [1,2]$$ ϱ ∈ [ 1 , 2 ] , we show that the associated solution semigroup S(t) is analytic and exponentially stable as well.

We provide a proof via direct energy estimates of the optimal exponential decay rate of the semigroup generated by the weakly damped wave equation.

We consider the nonlinear integrodifferential Benjamin-Bona-Mahony equation $$ u_t - u_{txx} + u_x - \int_0^\infty g(s) u_{xx}(t-s) {\rm d} s + u u_x = f $$ where the dissipation is entirely contributed by the memory term. Under a suitable smallness assumption on the external force $f$, we show that the related solution semigroup possesses the global attractor in the natural weak energy space. The result is obtained by means of a nonstandard approach based on the construction of a suitable family of attractors on certain invariant sets of the phase space.

We consider the nonclassical diffusion equation with hereditary memory \begin{document}$ u_t-\Delta u_t -\int_0^\infty \kappa(s)\Delta u(t-s)\,{{\rm{d}}} s +f(u) = g $\end{document} on a bounded three-dimensional domain. The main feature of the model is that the equation does not contain a term of the form \begin{document}$ -\Delta u $\end{document} , contributing as an instantaneous damping. Setting the problem in the past history framework, we prove that the related solution semigroup possesses a global attractor of optimal regularity.

We consider the Moore–Gibson–Thompson equation with memory of type II $$\begin{aligned} \partial _{ttt} u(t) + \alpha \partial _{tt} u(t) + \beta A \partial _t u(t) + \gamma Au(t)-\int _0^t g(t-s) A \partial _t u(s)\mathrm{d}s=0 \end{aligned}$$ where A is a strictly positive selfadjoint linear operator (bounded or unbounded) and $$\alpha ,\beta ,\gamma >0$$ satisfy the relation $$\gamma \le \alpha \beta $$ . First, we prove well-posedness of finite energy solutions, without requiring any restriction on the total mass $$\varrho $$ of g. This extends previous results in the literature, where such a restriction was imposed. Second, we address an open question within the context of longtime behavior of solutions. We show that an “overdamping” in the memory term can destabilize the originally stable dynamics. In fact, it is always possible to find memory kernels g, complying with the usual mass restriction $$\varrho

For ν∈(0,1), we investigate the nonlinear integro‐differential equation on a multidimensional space domain Ω⊂RnDtνu−L1u−∫0tK(t−s)L2u(·,s)ds=f(x,t,u)+g(x,t)where Dtν is the Caputo fractional derivative and L1 and L2 are uniform elliptic operators with smooth coefficients depending on time. Under suitable conditions on the nonlinearity, the global existence and uniqueness of the classical solution to the related initial and boundary value problems are established.

We consider the MGT equation with memory $$\partial_{ttt} u + \alpha \partial_{tt} u - \beta \Delta \partial_{t} u - \gamma\Delta u + \int_{0}^{t}g(s) \Delta u(t-s) ds = 0.$$ We prove an existence and uniqueness result removing the convexity assumption on the convolution kernel $g$, usually adopted in the literature. In the subcritical case $\alpha\beta>\gamma$, we establish the exponential decay of the energy, without leaning on the classical differential inequality involving $g$ and its derivative $g'$, namely, $$g'+\delta g\leq 0,\quad\delta>0,$$ but only asking that $g$ vanishes exponentially fast.

We study the energy transfer in the linear system \begin{document}$ \begin{cases} \ddot u+u+\dot u = b\dot v\\ \ddot v+v-\epsilon \dot v = -b\dot u \end{cases} $\end{document} made by two coupled differential equations, the first one dissipative and the second one antidissipative. We see how the competition between the damping and the antidamping mechanisms affect the whole system, depending on the coupling parameter \begin{document}$ b $\end{document} .

The contraction semigroup $$S(t)=\mathrm{e}^{t\mathbb {A}}$$ generated by the abstract linear dissipative evolution equation $$\begin{aligned} \ddot{u} + A u + f(A) \dot{u}=0 \end{aligned}$$ is analyzed, where A is a strictly positive selfadjoint operator and f is an arbitrary nonnegative continuous function on the spectrum of A. A full description of the spectrum of the infinitesimal generator $$\mathbb {A}$$ of S(t) is provided. Necessary and sufficient conditions for the stability, the semiuniform stability and the exponential stability of the semigroup are found, depending on the behavior of f and the spectral properties of its zero-set. Applications to wave, beam and plate equations with fractional damping are also discussed.

The aim of this note is to propose a mathematical model to describe aging effects occurring in viscoelastic materials. In the classical formulation, viscoelasticity is modeled through an integro-differential equation, where the convolution kernel takes into account the delay effects induced by viscosity. Usually, such a kernel is a given function, related to the rheological properties of the material. On the other hand, it is reasonable to conjecture that, as the material changes/deteriorates over time (and these changes are referred to as aging), the shape of the kernel should change as well. The key idea is then to consider an integro-differential equation where the convolution kernel is a function of time itself. This allows a more realistic description of the evolution, requiring the development of a new mathematical theory apt to treat dynamical systems acting on time-dependent spaces.

We address the energy transfer in the differential system { u t t t + α u t t − β Δ u t − γ Δ u = − η Δ θ θ t − κ Δ θ = η Δ u t t + α η Δ u t made by a Moore-Gibson-Thompson equation in the supercritical regime, hence antidissipative, coupled with the classical heat equation. The asymptotic properties of the related solution semigroup depend on the strength of the coupling, ruling the competition between the Fourier damping and the MGT antidamping. Exponential stability will be shown always to occur, provided that the coupling constant is sufficiently large with respect to the other structural parameters. A fact of general interest will be also discussed, namely, the impossibility of attaining the optimal exponential decay rate of a given dissipative system via energy estimates.

PurposeThe authors propose a rather elementary method to compute a family of integrals on the half line, involving positive powers of sin x and negative powers of x, depending on the integer parameters n≥q≥1.Design/methodology/approachCombinatorics, sine and cosine integral functions.FindingsThe authors prove an explicit formula to evaluate sinc-type integrals.Originality/valueThe proof is not present in the current literature, and it could be of interest for a large audience.

We consider a viscoelastic plate equation of Moore-Gibson-Thompson type coupled with two different kinds of thermal laws, namely, the usual Fourier one and the heat conduction law of type III. In both cases, the resulting system is shown to generate a contraction semigroup of solutions on a suitable Hilbert space. Then we prove that these semigroups are analytic, despite the fact that the semigroup generated by the mechanical equation alone does not share the same property. This means that the coupling with the heat equation produces a regularizing effect on the dynamics, implying in particular the impossibility of the localization of solutions. As a byproduct of our main result, the exponential stability of the semigroups is established.

We introduce a new mathematical framework for the time discretization of evolution equations with memory. As a model, we focus on an abstract version of the equation \begin{document}$ \partial_t u(t) - \int_0^\infty g(s) \Delta u(t-s)\, {{\rm{d}}} s = 0 $\end{document} with Dirichlet boundary conditions, modeling hereditary heat conduction with Gurtin-Pipkin thermal law. Well-posedness and exponential stability of the discrete scheme are shown, as well as the convergence to the solutions of the continuous problem when the time-step parameter vanishes.

In the present correction we add the missing sections 7, 8 and 9 to the original paper [1]. Such sections were present and peer-reviewed in the original submission, but they were mistakenly omitted during the preparation of the final version with the AIMS template.

For ν ∈ ( 0 , 1 ) , we consider the semilinear integro-differential equation on the one-dimensional domain Ω = ( a , b ) in the unknown u = u ( x , t ) D t ν u − L 1 u − ∫ 0 t K ( t − s ) L 2 u ( ⋅ , s ) d s = f ( x , t , u ) + g ( x , t ) where D t ν is the Caputo fractional derivative and L 1 and L 2 are uniform elliptic operators with time-dependent smooth coefficients. Under certain structural conditions on the nonlinearity f , the global existence and uniqueness of classical solutions to the related initial–boundary value problems are established, via the so-called continuation argument approach. The key point is looking for suitable a priori estimates of the solution in the fractional Holder spaces.

Given ρ ∈ [ 0 , 1 ] , we consider for e ∈ ( 0 , 1 ] the nonautonomous viscoelastic equation with a singularly oscillating external force ∂ t t u − κ ( 0 ) Δ u − ∫ 0 ∞ κ ′ ( s ) Δ u ( t − s ) d s + f ( u ) = g 0 ( t ) + e − ρ g 1 ( t / e ) together with the averaged equation ∂ t t u − κ ( 0 ) Δ u − ∫ 0 ∞ κ ′ ( s ) Δ u ( t − s ) d s + f ( u ) = g 0 ( t ) . Under suitable assumptions on the nonlinearity and on the external force, the related solution processes S e ( t , τ ) acting on the natural weak energy space H are shown to possess uniform attractors A e . Within the further assumption ρ 1 , the family A e turns out to be bounded in H , uniformly with respect to e ∈ [ 0 , 1 ] . The convergence of the attractors A e to the attractor A 0 of the averaged equation as e → 0 is also established.

An abstract version of the fourth-order equation $$\partial_{tttt}u+\alpha\partial_{ttt}u+\beta\partial_{tt}u-\gamma\Delta\partial_{tt}u-\delta\Delta\partial_{t}u-\varrho\Delta u=0$$ subject to the homogeneous Dirichlet boundary condition is analyzed. Such a model encompasses the Moore–Gibson–Thompson equation with memory in presence of an exponential kernel. The stability properties of the related solution semigroup are investigated. In particular, a necessary and sufficient condition for exponential stability is established, in terms of the values of certain stability numbers depending on the strictly positive parameters $${\alpha, \beta, \gamma, \delta, \varrho}$$ .

For ν ∈ ( 0 , 1 ) , we analyze the system of equations on the two dimensional domain Ω in the unknown function u = ( u 1 ( x , t ) , u 2 ( x , t ) ) u t − μ Δ u − γ Δ ∂ t ν u + u ⋅ ∇ u = − ∇ p + f , div u = 0 , where ∂ t ν denotes the Riemann–Liouville fractional derivative of order ν . Under suitable conditions on the given data, the global existence and uniqueness of strong solutions to the related initial–boundary value problems are established for ν ∈ ( 0 , 1 ∕ 2 ) . The convergence of the strong solution to the one of the corresponding initial–boundary value problem to the Navier–Stokes equations is discussed, in the limit ν → 0 . We also present several numerical tests illustrating this convergence.

We consider a linear Volterra integro-differential equation of hyperbolic type, which can be viewed as an abstract version of the equation $$\begin{aligned} \partial _{tt} u(t)- \varDelta u(t) +\displaystyle \int \limits _0^t\mu (s)\varDelta u(t-s)\mathrm{d}s=0 \end{aligned}$$describing the motion of linearly viscoelastic solids. We establish some decay results for the associated energy, under assumptions that do not involve differential inequalities for the convolution kernel $$\mu $$.

Further generalizations of the BCP Theorem 1.1 can be found in the literature (see, e.g., [30, 36]). We report here a couple of results in that direction.

In order to prove the main result of this chapter, the Schauder-Tychonoff fixed point theorem, we first need a definition and a lemma.

Let \((V,\langle \cdot ,\cdot \rangle _V,\Vert \cdot \Vert _V)\) be a real Hilbert space. Again, we denote the action of an element \(v^*\) of the dual space \(V^*\) on \(v\in V\) by \(\langle v^*, v\rangle \).

If T is a non-expansive linear map of a uniformly convex Banach space, then all the fixed points of T are recovered by means of a limit procedure.

Let X, Y be Banach spaces, let \(x_0\in U\subset X\) with U open, and let \(f:U\rightarrow Y\).

Let X, Y be Banach spaces, and \(f: B_X(x_0,r)\rightarrow Y\) be a Frechet differentiable map. In order to find a zero for f, the idea is to apply an iterative method constructing a sequence \(x_n\) starting from \(x_0\) so that \(x_{n+1}\) is the zero of the tangent \(\tau \) of f at the point \(x_{n}\) (see Definition 14.2). Assuming then \([f'(x)]^{-1}\in L(Y, X)\) on \(B_X(x_0,r)\), one has

In concrete applications it is somehow easier to work with functions defined on the whole space X, and rather ask more restrictive conditions, such as compactness, on the maps.

Let (X, d) be a complete metric space. Selecting an arbitrary \(x_0\in X\), that we may call the zero of X, we denote.

In this case, it is readily seen that there exists the smallest value \(\lambda \) for which the inequality holds, called the Lipschitz constant of f.