arXiv subjects
Publications and source records attributed to Vittorino Pata.
In this note we establish a unified growth rate for the operator norm of $C_0$-semigroups on Hilbert spaces whose generators satisfy the generalized Kreiss resolvent condition. Our bound contains and improves several known estimates in the literature. In particular, it captures the transition between different super-linear growth behaviors.
In this note we propose a definition of weak solution for an abstract Cauchy problem in a Hilbert space, and we discuss existence and uniqueness results.
The exponential decay rate of the semigroup $S(t)=e^{t\mathbb{A}}$ generated by the abstract damped wave equation $$\ddot u + 2f(A) \dot u +A u=0 $$ is here addressed, where $A$ is a strictly positive operator. The continuous function $f$, defined on the spectrum of $A$, is subject to the constraints $$\inf f(s)>0\qquad\text{and}\qquad \sup f(s)/s <\infty$$ which are known to be necessary and sufficient for exponential stability to occur. We prove that the operator norm of the semigroup fulfills the estimate $$\|S(t)\|\leq Ce^{σ_*t}$$ being $σ_*<0$ the supremum of the real part of the spectrum of $\mathbb{A}$. This estimate always holds except in the resonant cases, where the negative exponential $e^{σ_*t}$ turns out to be penalized by a factor $(1+t)$. The decay rate is the best possible allowed by the theory.
For $0<ν_2<ν_1\leq 1$, we analyze a linear integro-differential equation on the space-time cylinder $Ω\times(0,T)$ in the unknown $u=u(x,t)$ $$\mathbf{D}_{t}^{ν_1}(\varrho_{1}u)-\mathbf{D}_{t}^{ν_2}(\varrho_2 u)-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u =f$$ where $\mathbf{D}_{t}^{ν_i}$ are the Caputo fractional derivatives, $\varrho_i=\varrho_i(x,t)$ with $\varrho_1\geq μ_0>0$, $\mathcal{L}_{i}$ are uniform elliptic operators with time-dependent smooth coefficients, $\mathcal{K}$ is a summable convolution kernel, and $f$ is an external force. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under suitable conditions on the given data, the global classical solvability of the associated initial-boundary value problems is addressed. To this end, a special technique is needed, adapting the concept of a regularizer from the theory of parabolic equations. This allows us to remove the usual assumption about the nonnegativity of the kernel representing fractional derivatives. The problem is also investigated from the numerical point of view.
The purpose of this work is to produce a family of equations describing the evolution of the temperature in a rigid heat conductor. This is obtained by means of successive approximations of the Fourier law, via memory relaxations and integral perturbations.
We address the energy transfer in the differential system $$ \begin{cases} u_{ttt}+αu_{tt} - βΔu_t - γΔu = -ηΔθ\\ θ_t - κΔθ=ηΔu_{tt}+ αηΔu_t \end{cases} $$ 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.
We study the energy transfer in the linear system $$ \begin{cases} \ddot u+u+\dot u=b\dot v\\ \ddot v+v-ε\dot v=-b\dot u \end{cases} $$ 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 $b$.
We consider the MGT equation with memory $$\partial_{ttt} u + α\partial_{tt} u - βΔ\partial_{t} u - γΔu + \int_{0}^{t}g(s) Δu(t-s) ds = 0.$$ 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 $αβ>γ$, we establish the exponential decay of the energy, without leaning on the classical differential inequality involving $g$ and its derivative $g'$, namely, $$g'+δg\leq 0,\quadδ>0,$$ but only asking that $g$ vanishes exponentially fast.
We propose a rather elementary method to compute a certain family of integrals on the half line, depending on the integer parameters $n\geq q\geq 1$.
We consider the Moore-Gibson-Thompson equation with memory of type II $$ \partial_{ttt} u(t) + α\partial_{tt} u(t) + βA \partial_t u(t) + γAu(t)-\int_0^t g(t-s) A \partial_t u(s){\rm d} s=0 $$ where $A$ is a strictly positive selfadjoint linear operator (bounded or unbounded) and $α,β,γ>0$ satisfy the relation $γ\leqαβ$. First, we prove a well-posedness result without requiring any restriction on the total mass $\varrho$ of $g$. Then we show that it is always possible to find memory kernels $g$, complying with the usual mass restriction $\varrho<β$, such that the equation admits solutions with energy growing exponentially fast. In particular, this provides the answer to a question raised in "F. Dell'Oro, I. Lasiecka, V. Pata, J. Differential Equations 261 (2016), 4188-4222".
The contraction semigroup $S(t)={\rm e}^{t\mathbb{A}}$ generated by the abstract linear dissipative evolution equation $$ \ddot u + A u + f(A) \dot u=0 $$ 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.
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 a Navier-Stokes-Voigt fluid model where the instantaneous kinematic viscosity has been completely replaced by a memory term incorporating hereditary effects, in presence of Ekman damping. The dissipative character of our model is weaker than the one where hereditary and instantaneous viscosity coexist, previously studied by Gal and Tachim-Medjo. Nevertheless, we prove the existence of a regular exponential attractor of finite fractal dimension under rather sharp assumptions on the memory kernel.
Given $β\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version of the nonlinear stationary model in dimensionless form $$\begin{cases} u"" - \Big(β+ \varrho\int_0^1 |u'(s)|^2\,{\rm d} s\Big)u" +k(u-v) = 0 v"" - \Big(β+ \varrho\int_0^1 |v'(s)|^2\,{\rm d} s\Big)v" -k(u-v) = 0 \end{cases} $$ describing the equilibria of an elastically-coupled extensible double-beam system subject to evenly compressive axial loads. Necessary and sufficient conditions in order to have nontrivial solutions are established, and their explicit closed-form expressions are found. In particular, the solutions are shown to exhibit at most three nonvanishing Fourier modes. In spite of the symmetry of the system, nonsymmetric solutions appear, as well as solutions for which the elastic energy fails to be evenly distributed. Such a feature turns out to be of some relevance in the analysis of the longterm dynamics, for it may lead up to nonsymmetric energy exchanges between the two beams, mimicking the transition from vertical to torsional oscillations.
Given $ρ\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$ \partial_{tt} u-κ(0)Δu - \int_0^\infty κ'(s)Δu(t-s) d s +f(u)=g_{0}(t)+\varepsilon ^{-ρ}g_{1}(t/\varepsilon ) $$ together with the {\it averaged} equation $$ \partial_{tt} u-κ(0)Δu - \int_0^\infty κ'(s)Δu(t-s) d s +f(u)=g_{0}(t). $$ Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\varepsilon(t,τ)$ acting on the natural weak energy space ${\mathcal H}$ are shown to possess uniform attractors ${\mathcal A}^\varepsilon$. Within the further assumption $ρ<1$, the family ${\mathcal A}^\varepsilon$ turns out to be bounded in ${\mathcal H}$, uniformly with respect to $\varepsilon\in[0,1]$. The convergence of the attractors ${\mathcal A}^\varepsilon$ to the attractor ${\mathcal A}^0$ of the averaged equation as $\varepsilon\to 0$ is also established.
An elementary proof of a quantitative version of the Riemann-Lebesgue lemma for functions supported on the half line is given. Applications to differential models with memory are discussed.
We continue the analysis on the model equation arising in the theory of viscoelasticity $$ \partial_{tt} u(t)-\big[1+k_t(0)\big]Δu(t) -\int_0^\infty k'_t(s)Δu(t-s) d s + f(u(t)) = g $$ in the presence of a (convex, nonnegative and summable) memory kernel $k_t(\cdot)$ explicitly depending on time. Such a model is apt to describe, for instance, the dynamics of aging viscoelastic materials. The earlier paper [4] was concerned with the correct mathematical setting of the problem, and provided a well-posedness result within the novel theory of dynamical systems acting on time-dependent spaces, recently established by Di Plinio {\it et al.}\ [14] In this second work, we focus on the asymptotic properties of the solutions, proving the existence and the regularity of the time-dependent global attractor for the dynamical process generated by the equation. In addition, when $k_t$ approaches a multiple $mδ_0$ of the Dirac mass at zero as $t\to\infty$, we show that the asymptotic dynamics of our problem is close to the one of its formal limit $$\partial_{tt} u(t)-Δu(t) -mΔ\partial_t u(t)+ f(u(t)) = g$$ describing viscoelastic solids of Kelvin-Voigt type.
We consider the model equation arising in the theory of viscoelasticity $$\partial_{tt} u-h_t(0)Δu -\int_{0}^\infty h_t'(s)Δu(t-s)d s+ f(u) = g.$$ Here, the main feature is that the memory kernel $h_t(\cdot)$ depends on time, allowing for instance to describe the dynamics of aging materials. From the mathematical viewpoint, this translates into the study of dynamical systems acting on time-dependent spaces, according to the newly established theory of Di Plinio et al. In this first work, we give a proper notion of solution, and we provide a global well-posedness result. The techniques naturally extend to the analysis of the longterm behavior of the associated process, and can be exported to cover the case of general systems with memory in presence of time-dependent kernels.