Global attractors for a three-dimensional conserved phase-field system with memory
Gianluca Mola
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 $.
keywords: global attractor. Conserved phase-field models memory effects absorbing sets
Identification of a real constant in linear evolution equations in Hilbert spaces
Alfredo Lorenzi Gianluca Mola
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.
keywords: linear evolution equations in Hilbert spaces linear parabolic equations unknown constants well-posedness results Faedo-Galerkin approximation. Identification problems
Recovering a large number of diffusion constants in a parabolic equation from energy measurements
Gianluca Mola
$\left(H, \left\langle { \cdot , \cdot } \right\rangle \right)$
be a separable Hilbert space and
$A_{i}:D(A_i) \to H$
$i = 1,···,n$
) be a family of nonnegative and self-adjoint operators mutually commuting. We study the inverse problem consisting in the identification of a function
$u:[0,T] \to H$
$α_{1},···,α_{n} > 0$
(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
$\left\langle A_{1} u(T),u(T)\right\rangle = \varphi_{1}, ~~~··· ~~~,\left\langle A_{n} u(T),u(T)\right\rangle = \varphi_{n},$
are given positive constants. Under suitable assumptions on the operators
and on the initial data
$x ∈ H$
, 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.
keywords: Identification problems unknown constants linear evolution equations in Hilbert spaces linear parabolic equations well-posedness results
Semigroup-theoretic approach to identification of linear diffusion coefficients
Gianluca Mola Noboru Okazawa Jan Prüss Tomomi Yokota
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).
keywords: linear evolution equations in Banach spaces linear parabolic equations Identification problems unknown constants well-posedness results.

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