217 research outputs found
Long-Time Behavior of Quasilinear Thermoelastic Kirchhoff-Love Plates with Second Sound
We consider an initial-boundary-value problem for a thermoelastic Kirchhoff &
Love plate, thermally insulated and simply supported on the boundary,
incorporating rotational inertia and a quasilinear hypoelastic response, while
the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law
giving rise to a 'second sound' effect. We study the local well-posedness of
the resulting quasilinear mixed-order hyperbolic system in a suitable solution
class of smooth functions mapping into Sobolev -spaces. Exploiting the
sole source of energy dissipation entering the system through the hyperbolic
heat flux moment, provided the initial data are small in a lower topology
(basic energy level corresponding to weak solutions), we prove a nonlinear
stabilizability estimate furnishing global existence & uniqueness and
exponential decay of classical solutions.Comment: 46 page
EXPONENTIAL GROWTH OF SOLUTIONS FOR A VARIABLE-EXPONENT FOURTH-ORDER VISCOELASTIC EQUATION WITH NONLINEAR BOUNDARY FEEDBACK
In this paper we study a variable-exponent fourth-order viscoelastic equation of the formin a bounded domain of . Under suitable conditions on variable exponents and initial data, we prove that the solutions will grow up as an exponential function with positive initial energy level. Our result improves and extends many earlier results in the literature such as the on by Mahdi and Hakem (Ser. Math. Inform. 2020, https://doi.org/10.22190/FUMI2003647M)
Boundary Stabilization of Quasilinear Maxwell Equations
We investigate an initial-boundary value problem for a quasilinear
nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary
condition of Silver & M\"uller type in a smooth, bounded, strictly star-shaped
domain of . Imposing usual smallness assumptions in addition to
standard regularity and compatibility conditions, a nonlinear stabilizability
inequality is obtained by showing nonlinear dissipativity and
observability-like estimates enhanced by an intricate regularity analysis. With
the stabilizability inequality at hand, the classic nonlinear barrier method is
employed to prove that small initial data admit unique classical solutions that
exist globally and decay to zero at an exponential rate. Our approach is based
on a recently established local well-posedness theory in a class of
-valued functions.Comment: 22 page
A note on a class of 4th order hyperbolic problems with weak and strong damping and superlinear source term
In this paper we study a initial-boundary value problem for 4th order hyperbolic equations with weak and strong damping terms and superlinear source term. For blow-up solutions a lower bound of the blow-up time is derived. Then we extend the results to a class of equations where a positive power of gradient term is introduced
A global attractor for a fluid--plate interaction model accounting only for longitudinal deformations of the plate
We study asymptotic dynamics of a coupled system consisting of linearized 3D
Navier--Stokes equations in a bounded domain and the classical (nonlinear)
elastic plate equation for in-plane motions on a flexible flat part of the
boundary. The main peculiarity of the model is the assumption that the
transversal displacements of the plate are negligible relative to in-plane
displacements. This kind of models arises in the study of blood flows in large
arteries. Our main result states the existence of a compact global attractor of
finite dimension. We also show that the corresponding linearized system
generates exponentially stable -semigroup. We do not assume any kind of
mechanical damping in the plate component. Thus our results means that
dissipation of the energy in the fluid due to viscosity is sufficient to
stabilize the system.Comment: 18 page
Evolution Semigroups in Supersonic Flow-Plate Interactions
We consider the well-posedness of a model for a flow-structure interaction.
This model describes the dynamics of an elastic flexible plate with clamped
boundary conditions immersed in a supersonic flow. A perturbed wave equation
describes the flow potential. The plate's out-of-plane displacement can be
modeled by various nonlinear plate equations (including von Karman and Berger).
We show that the linearized model is well-posed on the state space (as given by
finite energy considerations) and generates a strongly continuous semigroup. We
make use of these results to conclude global-in-time well-posedness for the
fully nonlinear model.
The proof of generation has two novel features, namely: (1) we introduce a
new flow potential velocity-type variable which makes it possible to cover both
subsonic and supersonic cases, and to split the dynamics generating operator
into a skew-adjoint component and a perturbation acting outside of the state
space. Performing semigroup analysis also requires a nontrivial approximation
of the domain of the generator. And (2) we make critical use of hidden
regularity for the flow component of the model (in the abstract setup for the
semigroup problem) which allows us run a fixed point argument and eventually
conclude well-posedness. This well-posedness result for supersonic flows (in
the absence of rotational inertia) has been hereto open. The use of semigroup
methods to obtain well-posedness opens this model to long-time behavior
considerations.Comment: 31 page
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