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### Open Access Journals

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

be a separable Hilbert space and

(

) be a family of nonnegative and self-adjoint operators mutually commuting. We study the inverse problem consisting in the identification of a function

and

constants

(

$\left(H, \left\langle { \cdot , \cdot } \right\rangle \right)$ |

$A_{i}:D(A_i) \to H$ |

$i = 1,···,n$ |

$u:[0,T] \to H$ |

$n$ |

$α_{1},···,α_{n} > 0$ |

*diffusion coefficients*) that fulfill the initial-value problem$ u'(t) + α_{1} A_{1}u(t) + ··· + α_{n} A_{n}u(t) = 0, ~~~t ∈ (0,T), ~~~u(0) = x,$ |

and the additional conditions

$\left\langle A_{1} u(T),u(T)\right\rangle = \varphi_{1}, ~~~··· ~~~,\left\langle A_{n} u(T),u(T)\right\rangle = \varphi_{n},$ |

where

are given positive constants. Under suitable assumptions on the operators

and on the initial data

, we shall prove that the solution of such a problem is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion constants in a heat equation and of the Lamé parameters in a elasticity problem on a plate.

$\varphi_{i}$ |

$A_{i}$ |

$x ∈ H$ |

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.

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