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CPAA

We consider a conserved phase-field system on a tridimensional
bounded domain. The heat conduction is characterized by memory
effects depending on the past history of the (relative) temperature
$\vartheta$. These effects are represented through a convolution
integral whose relaxation kernel $k$ is a summable and decreasing
function. Therefore the system consists of a linear
integrodifferential equation for $\vartheta$ which is coupled with a
viscous Cahn-Hilliard type equation governing the order parameter
$\chi$. The latter equation contains a nonmonotone nonlinearity
$\phi$ and the viscosity effects are taken into account by the term
$-\alpha \Delta\chi_t$, for some $\alpha \geq 0$. Thus, we formulate a
Cauchy-Neumann problem depending on $\alpha $. Assuming suitable
conditions on $k$, we prove that this problem generates a
dissipative strongly continuous semigroup $S^\alpha (t)$ on an
appropriate phase space accounting for the past histories of
$\vartheta$ as well as for the conservation of the spatial means of
the enthalpy $\vartheta+\chi$ and of the order parameter. We first
show, for any $\alpha \geq 0$, the existence of the global attractor
$\mathcal A_\alpha $. Also, in the viscous case ($\alpha > 0$), we prove the
finiteness of the fractal dimension and the smoothness of
$\mathcal A_\alpha $.

IPI

Let $H$ be a real separable Hilbert space and $A:\mathcal{D}(A) \to H$ be a positive and self-adjoint (unbounded) operator, and denote by $A^\sigma$ its power of exponent $\sigma \in [-1,1)$. We consider the identification problem consisting in searching for a function $u:[0,T] \to H$ and a real constant $\mu$ that fulfill the initial-value problem
$$
u' + Au = \mu \, A^\sigma u, \quad t \in (0,T), \quad u(0) = u_0,
$$
and the additional condition
$$
\alpha \|u(T)\|^{2} + \beta \int_{0}^{T}\|A^{1/2}u(\tau)\|^{2}d\tau = \rho,
$$
where $u_{0} \in H$, $u_{0} \neq 0$ and $\alpha, \beta \geq 0$, $\alpha+\beta > 0$ and $\rho >0$ are given. By means of a finite-dimensional approximation scheme, we construct a unique solution $(u,\mu)$ of suitable regularity on the whole interval $[0,T]$, and exhibit an explicit continuous dependence estimate of Lipschitz-type with respect to the data $u_{0}$ and $\rho $. Also, we provide specific applications to second and fourth-order parabolic initial-boundary value problems.

DCDS-S

Let $X$ be a complex Banach space and
$A:\,D(A) \to X$ a quasi-$m$-sectorial operator
in $X$. This paper is concerned with the
identification of diffusion coefficients
$\nu > 0$ in the initial-value problem:
\[
(d/dt)u(t) + {\nu}Au(t) = 0,
\quad t \in (0,T), \quad u(0) = x \in X,
\]
with additional condition $\|u(T)\| = \rho$,
where $\rho >0$ is known. Except for
the additional condition, the solution to the
initial-value problem is given by
$u(t) := e^{-t\,{\nu}A} x
\in C([0,T];X) \cap C^{1}((0,T];X)$.
Therefore, the identification of $\nu$ is reduced
to solving the equation
$\|e^{-{\nu}TA}x\| = \rho$.
It will be shown that the unique root
$\nu = \nu(x,\rho)$
depends on $(x,\rho)$ locally Lipschitz
continuously if the datum $(x,\rho)$ fulfills
the restriction $\|x\|> \rho$. This extends
those results in
Mola [6](2011).

## Year of publication

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