IDENTIFICATION OF A SOURCE TERM AND A COEFFICIENT IN A PARABOLIC DEGENERATE PROBLEM
Tesi di Dottorato
Data di Pubblicazione:
2012
Citazione:
IDENTIFICATION OF A SOURCE TERM AND A COEFFICIENT IN A PARABOLIC DEGENERATE PROBLEM / U. Fedus ; tutor: A. Lorenzi ; coordinatore: M. Peloso. Universita' degli Studi di Milano, 2012 Feb 20. 24. ciclo, Anno Accademico 2011. [10.13130/fedus-ulyana_phd2012-02-20].
Abstract:
The globally in time existence and uniqueness of solutions to inverse problems is one of the most difficult questions to be answered.
Even though the direct problems are well-posed in the sense of Hadamard (i.e. existence, uniqueness and stability results hold true),
the inverse ones generally are not. The situation gets more complicated if
the equation contains more than one unknown coefficient, and even more if
the unknown functions depend on different variables.
We consider the following
identification abstract problem in a general Banach space $X$: find a
function $u:[0,T] \to X,$ a coefficient $a_1:[0,T] \to \mathbb R$ and a vector $z \in X$ such that the
initial-value problem
\begin{align}
&\frac{1}{a_0(t)}\ u'(t)-Au(t)-a_1(t)u(t)\!=\!f(t)z+g(t), \qq u(0)=u_0 \label{zi2}
\end{align}
is fulfilled, where $a_0(t)>0$ and $a_0(t)=0$ only in some negligible set,
while $A:D(A)\subset X \to X$ is a
closed linear operator, $f$ is scalar functions and $g$ is a $X$-valued source
term.
The occurrence of two unknowns require to introduce two additional
conditions. We choose the first as nonlocal one in the integral form $\imi \!\!\varphi(t)u(t)d\mu(t)\!=\!h,$
where $\mu$ is a Borel measure on the interval $[0,T].$
The latter is of the following form:
$\Phi[u(t)]=k(t), \: t\!\in\! [0,T],$ where $\Phi$ is a prescribed linear continuous functional.
Here the functions $h$, $k, \varphi$ are scalar.
So, we investigate the problem (\ref{zi2})
along with these additional conditions. We study explicitly the case
of the \textit{Dirac measure} concentrated at $t=T_1, 0
Tipologia IRIS:
Tesi di dottorato
Elenco autori:
U. Fedus
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