91,385 research outputs found
Second Order Linear Evolution Equations with General Dissipation
No full textThe contraction semigroup S(t) = etA generated by the abstract linear dissipative evolution equation u ̈+Au+f(A)u ̇=0is 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 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
Robust exponential attractors for the strongly damped waveequation with memory. I
No full textWe consider the singular limit of the semilinear strongly damped wave equation with memory ∂ ttu - γΔ ∂ t u - k (0)Δ u - ∫0∞ {k'} (s)Δ u(t - s)ds + φ (u) = f, in presence of an arbitrarily growing nonlinearity φ, as the memory kernel k(s)-k(∞) converges to the Dirac mass at zero. The existence of a robust family of regular exponential attractors is established, under a necessary and sufficient condition on k, along with quantitative estimates of the closeness of the equation with memory to the corresponding limit equation. © 2008 Pleiades Publishing, Ltd
A hierarchy of heat conduction laws
No full textThe 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
On Pata–Suzuki-Type Contractions
No full textIn this manuscript, we introduce two notions, Pata–Suzuki Z -contraction and Pata Z -contraction for the pair of self-mapping g , f in the context of metric spaces. For such types of contractions, both the existence and uniqueness of a common fixed point are examined. We provide examples to illustrate the validity of the given results. Further, we consider ordinary differential equations to apply our obtained results
Some measurability and continuity properties of arbitrary real functions
No full textGiven an arbitrary real function f , the set D_f of all points where f admits approximate limit is the maximal (with respect to the relation of inclusion except for a nullset) measurable subset of the real line having the properties that the restriction of f to D_f is measurable, and f is approximately continuous at almost every point of D_f . These results extend the well-known fact that a function is measurable if and only if it is approximately continuous almost everywhere. In addition, there exists a maximal G_δ -set C_f (which can be actually constructed from f ) such that it is possible to find a functiong = f almost everywhere, whose set of points of continuity is exactly C_f .<br /
Long-term analysis of strongly damped nonlinear wave equations
No full textWe consider the strongly damped nonlinear wave equation
u_tt − Delta u_t − Delta u + f (u_t ) + g(u) = h
with Dirichlet boundary conditions, which serves as a model in the description
of thermal evolution within the theory of type III heat conduction. In particular,
the nonlinearity f acting on u_t is allowed to be nonmonotone and to exhibit a
critical growth of polynomial order 5. The main focus is the long-term analysis
of the related solution semigroup, which is shown to possess the global attractor
in the natural weak energy space
Delayed benign surgery during the COVID-19 pandemic. The other side of the coin
No full textnot availabl
Lack of superstable trajectories in linear viscoelasticity: a numerical approach
No full textGiven a positive operator on some Hilbert space,
and a nonnegative decreasing summable function ,
we consider the abstract equation with memory
modeling the dynamics of linearly viscoelastic solids.
The purpose of this work is to provide numerical evidence
of the fact that the energy
\E(t)=\Big(1-\int_0^t\mu(s)ds\Big)\|u(t)\|^2_1+\|\dot u(t)\|^2
+\int_0^t\mu(s)\|u(t)-u(t-s)\|^2_1ds,
of any nontrivial solution cannot decay faster than exponential,
no matter how fast might be the decay of the memory kernel .
This will be accomplished by simulating the integro-differential
equation for different choices of the memory kernel
and of the initial data
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