Related papers: On exponential decay rate of semigroup associated …

The Cauchy problem for second order linear differential equation $u''(t)+Du'(t)+Au(t)=0$ in Hilbert space $H$ with a sectorial operator $A$ and an accretive operator $D$ is studied. Sufficient conditions for exponential decay of the…

In this study, we analyze a semilinear damped evolution equation under different damping conditions, including the undamped $(\theta=0)$, effectively damped $(0<2\theta<\sigma)$, critically damped $(2\theta=\sigma)$, and non-effectively…

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 $f$,…

We present a new and simple bound for the exponential decay of second order systems using the spectral shift. This result is applied to finite matrices as well as to partial differential equations of Mathematical Physics. The type of the…

In this paper, we provide the spectral decomposition in Hilbert space of the $\mathcal{C}_0$-semigroup $P$ and its adjoint $\hatP$ having as generator, respectively, the Caputo and the right-sided Riemann-Liouville fractional derivatives of…

We study the Cauchy problem for the equation of the form $$ \ddot{u}(t) + (\aa A + B)\dot{u}(t) + (A+G)u(t) = 0,\tag* $$ where $A$, $B$, and $G$ are \o s in a Hilbert space $\Cal H$ with $A$ selfadjoint, $\sigma(A)=[0,\infty)$, $B\ge0$…

In this paper, we investigate the spectrum of the self adjoint differential operator with operator coefficitent in a separable Hilbert space. We also derive asymptotic formulas for the sum of eigenvalues of this operator.

We consider a class of differential-algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the…

Assume that $(X,d,\mu)$ is a metric space endowed with a non-negative Borel measure $\mu$ satisfying the doubling condition and the additional condition that $\mu(B(x,r))\gtrsim r^n$ for any $x\in X, \,r>0$ and some $n\geq1$. Let $L$ be a…

We consider an abstract second order evolution equation with damping. The "elastic" term is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We…

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…

We investigate rates of decay for $C_0$-semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to…

We study relations between the decaying rates of operator semigroups on Hilbert spaces and some spectral properties of their respective generators; in particular, we show that the decaying rates of orbits of semigroups which are stable but…

Let A be a positive self-adjoint linear operator acting on a real Hilbert space H and $\alpha$, c be positive constants. We show that all solutions of the evolution equation u + Au + cA $\alpha$ u = 0 with u(0) $\in$ D(A 1 2), u (0) $\in$ H…

In this paper we study in a Hilbert space a homogeneous linear second order difference equation with nonconstant and noncommuting operator coefficients. We build its exact resolutive formula consisting in the explicit non-iterative…

For a class of non-selfadjoint $h$--pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin.…

A lower semi-definite self-adjoint linear operator in a Hilbert space is taken whose discrete spectrum is not empty and comprises at least several eigenvalues $\lambda_{min}=\lambda_1\leqslant\ldots\leqslant\lambda_m<\sigma_{ess}$. The…

In this paper, we mainly discuss asymptotic profiles of solutions to a class of abstract second-order evolution equations of the form $u''+Au+u'=0$ in real Hilbert spaces, where $A$ is a nonnegative selfadjoint operator. The main result is…

We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel. Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as…

Let $T$ be a power-bounded linear operator on a Hilbert space $X$, and let $S$ be a bounded linear operator from another Hilbert space $Y$ to $X$. We investigate the non-exponential rate of decay of $\|T^nS\|$ as $n \to \infty$. First, when…