We are studying a class of nonlinear nonlocal diffusion problems associated with a p-Laplace-type operator, where a nonlocal quantity is present in the diffusion coefficient. We address the issues of existence and uniqueness for the parabolic setting. Then, we study the asymptotic behavior of the solution for large time. For this purpose, we introduce and investigate, in detail, the associated stationary problem. Moreover, since the solutions of the stationary problem are also critical points of some energy functional, we make a classification of its critical points.

We investigate the existence of multiple nontrivial solutions of a quasilinear elliptic Dirichlet problem depending on a parameter $\lambda>0$ of the form $$ -\Delta_pu=\lambda f(u)\quad\mbox{in }\ \Omega,\quad u=0\quad\mbox{on }\ \partial\Omega, $$ where $\Omega\subset \mathbb{R}^N$ is a bounded domain, $\Delta_p$, $1 < p < +\infty$, is the $p$-Laplacian, and $f: \mathbb{R}\to \mathbb{R}$ is a continuous function satisfying a subcritical growth condition. More precisely, we establish a variational approach that when combined with differential inequality techniques, allows us to explicitly describe intervals for the parameter $\lambda$ for which the problem under consideration admits nontrivial constant-sign as well as nodal (sign-changing) solutions. In our approach, a crucial role plays an abstract critical point result for functionals whose critical points are attained in certain open level sets. To the best of our knowledge, the novelty of this paper is twofold. First, neither an asymptotic condition for $f$ at zero nor at infinity is required to ensure multiple constant-sign solutions. Second, only by imposing some $\liminf$ and $\limsup$ condition of $f$ at zero, the existence of at least three nontrivial solutions including one nodal solution can be proved.

We consider a non-autonomous evolutionary problem \[ \dot{u} (t)+\mathcal A(t)u(t)=f(t), \quad u(0)=u_0 \] where the operator $\mathcal A(t)\colon V\to V^\prime$ is associated with a form $\mathfrak{a}(t,.,.)\colon V\times V \to \mathbb R$ and $u_0\in V$. Our main concern is to prove well-posedness with maximal regularity, which means the following. Given a Hilbert space $H$ such that $V$ is continuously and densely embedded into $H$ and given $f\in L^2(0,T;H)$, we are interested in solutions $u \in H^1(0,T;H)\cap L^2(0,T;V)$. We do prove well-posedness in this sense whenever the form is piecewise Lipschitz-continuous and satisfies the square root property. Moreover, we show that each solution is in $C([0,T];V)$. The results are applied to non-autonomous Robin-boundary conditions and maximal regularity is used to solve a quasilinear problem.

We are investigating regularity issues for parabolic almost minimizers of integral functionals with polynomial growth structure. In particular, we are interested in functions with values in the target space $\mathbb R^N$ for a dimension $N > 1$ and our aim is to prove Hölder continuity for the spatial gradient of an almost minimizer on an open set of full Lebesgue measure. As classical counter examples show, the regularity results which we achieve are optimal if we are dealing with general almost minimizers and dimensions of the target space strictly greater than one. However, the results are also valid in the case $N=1$, even if they are not optimal in that case. Our contribution can be seen as a natural generalization of the elliptic regularity theory for the gradient of vector--valued almost minimizers, which was developed in the 1980s to the framework of time-dependent, parabolic, problems. The parabolic nature of the problem, in combination with the fact that almost minimizers, in general, do not satisfy any associated parabolic partial differential equation, causes a variety of difficulties in regularity theory, which have to be overcome. The first part of the paper covers integral functionals, which involve merely the spatial gradient of the almost minimizing function. In this case, we obtain a partial Hölder regularity result for the spatial gradient with a Hölder exponent, which can be determined in a quantified way in dependence of the data of the problem. In the second part, we develop the theory for integrands, which involve not only the spatial gradient, but also the almost minimizer itself. In this more general case, the proof becomes substantially more involved, since a priori higher integrability of the spatial gradient has to be used. As a consequence, the explicit control on the Hölder continuity exponent gets lost. This is also in accordance with the elliptic theory where the same phenomenon occurs.

We study the feedback stabilization of a system composed by an incompressible viscous fluid and a rigid body. We stabilize the position and the velocity of the rigid body and the velocity of the fluid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the fluid domain and with values in a finite dimensional space. Our first result concerns weak solutions in the two-dimensional case, for initial data close to the stationary state. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain of the stationary state and of the stabilized solution are different. This additional difficulty leads to the assumption that the initial position of the rigid body is the position associated to the stationary state. Without this hypothesis, we work with strong solutions, and to deal with compatibility conditions at the initial time, we use finite dimensional dynamical controls. We again prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the rigid body and the velocity of the fluid. In the three dimensional case, we also obtain the local stabilization of strong solutions with finite dimensional dynamical controls.