IDENTIFICATION OF A SOURCE TERM AND A COEFFICIENT IN A PARABOLIC DEGENERATE PROBLEM

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 $t=T_1, 0<T_1\leq T$ and the one of an \textit{absolutely continuous measure $\mu.$} This thesis is devoted to investigation of inverse problems for degenerate parabolic equations aiming at the determination of one time-dependent coefficient $a_1$ and a spatial source term $z.$ So, the goal of this work is to find sufficient conditions on our data and operator $A$ under which the problem turns out to be well-posed. By means of Semigroup Theory and the Banach fixed-point theorem, we can find out sufficient conditions on the data $(f, g, u_0, h, k)$ ensuring \textit{global-in-time} existence and uniqueness for the solution $(a_1,u,z) \in L^1(0,T; {\mathbb R})\times \bigl[W^{1,1}(0,T;X) \cap L^\infty(0,T; D(A))\bigr]\times X.$ Moreover, a continuous dependence of Lipschitz type of the solution on the data is provided. We stress that we are obliged to introduce an unusual distance involving the data accounting for the degeneracy of function $a_0$. Finally, using a suitable metric for the data, we apply such results to a concrete parabolic problem.