Many fundamental biophysical processes, from cell division to cellular motility, involve dynamics of thin structures immersed in a very viscous fluid. Various popular models have been developed to describe this interaction mathematically, but much of our understanding of these models is only at the level of numerics and formal asymptotics. Here we seek to develop the PDE theory of filament hydrodynamics. 

We first propose a PDE framework for analyzing the error introduced by slender body theory (SBT), a common approximation used to facilitate computational simulations of immersed filaments in 3D. Given data prescribed only along a 1D curve, we develop a novel type of boundary value problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing.

We then consider other physically relevant scenarios in which the slender body PDE framework applies and shed light on the analysis of such problems.

In this lecture, we shall  present  the local null controllability of an initial boundary value problem for a thermistor equation. The control is distributed, locally in space. The main ingredients of the proof are suitable Carleman estimates for an adjoint system and Liusternik’s Inverse Mapping Theorem in Hilbert spaces. Also,  we  present   large time null controllability.

In a pioneering paper, on 1910, S. Zaremba established the \textit{strong maximum principle} saying that if $\Omega$ is a smooth bounded domain in $% \mathbb{R}^{N}$ and a function $u$ verifies \begin{equation}\left\{\begin{array}{lr} -\Delta u\geq f(x) & \text{in }\Omega , \\u=0 & \text{on }\partial \Omega ,%\end{array}%\right. \label{ProbLineal}\end{equation}% \noindent then \begin{equation} u(x)>0\text{ in }\Omega , \tag{P$_{u}$} \label{Positiv u} \end{equation} \noindent assumed that \begin{equation} f(x)\geq 0\text{ in }\Omega ,\text{ }f\neq 0. \tag{P$_{f}$} \label{Positiv f} \end{equation} The extension to a more general second order elliptic operator was due to E. Hopf, on 1927. Moreover, some years later, on 1952, E. Hopf and O.A. Oleinik, independently, proved that under the above conditions the normal derivative of $u$ satisfies the following sign condition \begin{equation} \frac{\partial u}{\partial n}<0\text{ on }\partial \Omega . \label{Hypo2.3} \end{equation}. The main goal of this lecture is to show that the non-changing sign assumption (\ref{Positiv f}) can be removed in a suitably way (for instance when $f(x)<0$ in some neighborhood of $\partial \Omega $)$.$ Under such type of new assumptions on $f$ we can prove that: (A) the positivity of $u,$ property (\ref{Positiv u}), still holds and $% \frac{\partial u}{\partial n}\leq 0$ on $\partial \Omega ,$ \noindent and (B) under additional conditions on $f(x)$, the positive \textit{solution} of the equation (\ref{ProbLineal}) [i.e., now with the identity symbol $=,$ instead $\geq $] does not satisfy (\ref{Hypo2.3}) but $\frac{\partial u}{% \partial n}=0$ on $\partial \Omega $ \ (this corresponds to the notion of \textit{flat solutions} already considered by different authors in the framework of some nonlinear problems). The above extension of the strong maximum principle admits many generalizations (nonlinear elliptic operators, zero and first order terms, etc.). In the case of linear parabolic problems%\[ \left\{\begin{array}{lr} u_{t}-\Delta u=f(x,t) & \text{in }\Omega \times (0,+\infty ), \\ u=0 & \text{on }\partial \Omega \times (0,+\infty ), \\ u(x,0)=u_{0}(x) & \text{on }\Omega ,% \end{array}% \right.\]% it can be proved that, under suitably changing sign conditions on $u_{0}(x)$ and/or on $f(x,t)$ (even if it occurs for any $t>0$), we get the global positivity of $u(x,t)$ on $\Omega $, for large values of time ( $\exists $ $% t_{0}\geq 0$ such that $u(x,t)>0$ a.e. $x\in \Omega ,$ for any $t>t_{0}$). An application to some \textit{sublinear} \textit{indefinite} semilinear equations will be also given. [Joint results with Jes\'{u}s Hern\'{a}ndez].

We analyze a class of semilinear damped wave-type equations with a delay feedback with time-variable time delay. Under suitable assumptions on the system’s parameters, by combining semigroup arguments, careful energy estimates, and an iterative approach, we are able to prove a well-posedness result and an exponential decay estimate for solutions corresponding to small initial data. An extension to viscoelastic wave equations with time delay is also discussed.