106,854 research outputs found
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
Global attractors for the extensible thermoelastic beam system
No full textAbstractThis work is focused on the dissipative system{∂ttu+∂xxxxu+∂xxθ−(β+‖∂xu‖L2(0,1)2)∂xxu=f,∂tθ−∂xxθ−∂xxtu=g describing the dynamics of an extensible thermoelastic beam, where the dissipation is entirely contributed by the second equation ruling the evolution of θ. Under natural boundary conditions, we prove the existence of bounded absorbing sets. When the external sources f and g are time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity for all parameters β∈R. The same result holds true when the first equation is replaced by∂ttu−γ∂xxttu+∂xxxxu+∂xxθ−(β+‖∂xu‖L2(0,1)2)∂xxu=f with γ>0. In both cases, the solutions on the attractor are strong solutions
On the Moore-Gibson-Thompson equation with thermal effects of Gurtin-Pipkin type
No full textWe consider the Moore-Gibson-Thompson-Gurtin-Pipkin model
\begin{cases}
u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u =- \varrho\Delta \theta \\
\noalign{\vskip1mm}
\displaystyle \theta_t - \int_0^\infty g(s)\Delta\theta(t-s)ds = \varrho\Delta u_{tt} + \varrho\alpha\Delta u_t
\end{cases}
with the first equation in the subcritical regime \alpha\beta>\gamma. The system generates
a strongly continuous semigroup of linear contractions which is never exponentially stable,
even if the second equation, when uncoupled, generates an exponentially stable semigroup.
This is in deep contrast to what happens in connection with the semigroup generated by the
Moore-Gibson-Thompson-Fourier system
\begin{cases}
u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u =- \varrho\Delta \theta \\
\noalign{\vskip1mm}
\displaystyle \theta_t - \nu\Delta\theta = \varrho\Delta u_{tt} + \varrho\alpha\Delta u_t
\end{cases}
formally obtained as a limit by letting
Delayed benign surgery during the COVID-19 pandemic. The other side of the coin
No full textnot availabl
On the Moore-Gibson-Thompson equation with memory with nonconvex kernels
No full textWe consider the MGT equation with memory
partial derivative(ttt)u + alpha partial derivative(tt)u - beta Delta partial derivative(t)u - gamma Delta u + integral(t)(0) 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 <= 0, delta > 0,
but we ask only that g vanish exponentially fast
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
Energy decay of electromagnetic systems with memory
No full textWe consider a linear evolution problem with memory arising in the theory of hereditary
electromagnetism. Under general assumptions on the memory kernel, all single trajectories
are proved to decay to zero, but the decay rate is not uniform in dependence of
the initial data, so that the system is not exponentially stable. Nonetheless, if the kernel
is rapidly fading and close to the Dirac mass at zero, then the solutions are close to
exponentially stable trajectories
Exponential stability in linear heat conduction with memory: a semigroup approach
No full textThis paper is concerned with the asymptotic behavior in time of solutions
to a linear problem arising in the theory of heat conduction with memory. In a
rigid heat conductor obeying the Gurtin-Pipkin constitutive model for the
heat flux, the energy balance leads to an hyperbolic, linear integro-differential
equation. In spite of the presence of a convolution term, the homogeneous
original problem, subject to initial-history conditions,
is transformed into an autonomous system by a suitable choice of variables. By
means of semigroup techniques the exponential decay of solutions is provided
- …
