The structure of positive solutions to nonlinear diffusion problems of the form $-\text{div}\ (|{\nabla} u|^{p-2}{\nabla} u) = \lambda f(u)$, in $\Omega$, $u = 0$ on $\partial \Omega$, $p > 1$, $\Omega \subset \mathbb R^N$ a bounded, smooth domain, is precisely studied as $\lambda \to +\infty $, for a class of logistic-type nonlinearities $f(u)$. By logistic it is understood that $f(u)/u^{p-1}$ is decreasing in $u > 0$, $f(u) \sim m u^{p-1}$, $m > 0$, as $u\to 0+$, while $f$ has a positive zero $u = u_0$ of order $k$. It is shown that the positive solution ${u_\lambda}$ homogenizes towards $u_0$ as $\lambda \to +\infty $, and develops a boundary layer near $\partial\Omega$ whose width is exactly measured. On the other hand, the arising of ``dead cores" $\{{u_\lambda} = u_0\}$ for $\lambda$ large is shown in the parameters regime $k < p-1$, the distance $\text{dist}(\{{u_\lambda} = u_0\},\partial\Omega)$ to $\partial \Omega$ being also exactly estimated as $\lambda \to +\infty $. Thus, earlier results in [12], [22] are substantially sharpened. In addition, suitable lower-order perturbations at infinity of the problem are studied.

We consider an initial and boundary value problem for a nonlinear Volterra integrodifferential equation. This equation governs the evolution of a pair of state variables, $u$ and $\vartheta$, which are mutually related by a maximal monotone graph $\gamma$ in ${{\Bbb R}}\times{{\Bbb R}}.$ The model can be viewed, for instance, as a generalized Stefan problem within the theory of heat conduction in materials with memory. Besides, it can be used for describing some diffusion processes in fractured media. The relation defined by $\gamma$ is properly interpreted and generalized in terms of a subdifferential operator associated with $\gamma$ and acting from $H^1(\Omega)$ to its dual space. Then, the generalized problem is formulated as an abstract Cauchy problem for a perturbation of a nonlinear semigroup, and existence and uniqueness of a solution $(u,\vartheta)$ can be proved via a fixed-point argument whatever the maximal monotone graph $\gamma$ is. Moreover, the meaning of $\gamma$ as a pointwise relationship is recovered almost everywhere, in the case when $\gamma$ is bounded on bounded subsets of ${{\Bbb R}}$. Finally, under some other restrictions on $\gamma$, the longtime behavior of the solution is investigated, in a more specific context related to the generalized Stefan problem.

We study the initial value problem for the KP-II equation. We prove the existence of solutions to the integral equation corresponding to KP-II for any data in $L^2$, removing the additional condition imposed in [15]. Following a method recently developed by J. Bourgain we obtain global solutions to KP-II with data below $L^2$.

We consider soliton solutions of the mean curvature flow, i.e., solutions which move under the mean curvature flow by a group of isometries of the ambient manifold. Several examples of solitons on manifolds are discussed. Moreover we present a local existence result for rotating solitons. We also prove global existence and stability for perturbed initial data close to a local soliton.

An initial boundary value problem for a thermoelastic plate is considered. Results about existence, uniqueness and asymptotic stability of the solutions are obtained as a consequence of the dissipation properties of the material. By using suitable multiplicative techniques we show the exponential decay rate of the energy.

We consider some free boundary problems involving motion by mean curvature and a nonlocal term. They arise as singular limits of various phase transition models. Using precise regularity estimates in Hölder spaces, we prove that these problems are well-posed. We study the qualitative behavior of the motion law and show in particular that the inclusion of interfaces is not preserved in time.

By using HUM we prove the exact boundary controllability of the transmission plate equations.

We consider real potentials $V$ such that the Schrödinger operator $-\Delta+V$ maps the Sobolev space $W^{2,p}(\mathbb{R}^{N})$ continuously into $L^{p}(\mathbb{R}^{N})$ for a range of values of $p$ which includes 2. Let $\sigma_{e}$ denote the essential spectrum of $-\Delta+V$ as a self-adjoint operator in $L^{2}(\mathbb{R}^{N}).$ If $\lambda\notin$ $\sigma_{e},$ we show that for all $p$ in the range considered, $-\Delta+V-\lambda:W^{2,p}% (\mathbb{R}^{N})\rightarrow L^{p}(\mathbb{R}^{N})$ is a Fredholm operator of index zero, that ker {$-\Delta+V-\lambda$\} is independent of $p$ and that $L^{p}(\mathbb{R}^{N})=$ker {$-\Delta+V-\lambda$\}$\oplus$\{$-\Delta +V-\lambda$\}$W^{2,p}(\mathbb{R}^{N}).$

In this paper we consider the existence, uniqueness and asymptotic behaviour of the solutions of nonlinear degenerate equations on manifolds. The existence is proved by making use of the Galerkin method and the uniform decay is obtained considering the perturbed energy method.

We consider the smallest viscosity solution of the Hessian equation $S_k(D^2u)=f(u)$ in $\omega$, that becomes infinite at the boundary of $\omega\subset\mathbb R^n$; here $S_k(D^2u)$ denotes the $k$-th elementary symmetric function of the eigenvalues of $D^2u$, for $k\in\{1,\dots, n\}$. We prove that if $\omega$ is strictly convex and $f$ satisfies suitable assumptions, then the smallest solution is convex. We also establish asymptotic estimates for the behaviour of such a solution near the boundary of $\omega$.

We prove the existence of connected branches of solutions having prescribed nodal properties for some boundary value problems depending on a parameter. The results are obtained via an elementary approach based on the shooting method and using a lemma from plane topology.

Let $u_t+f(u)_x=0$ be a strictly hyperbolic, genuinely nonlinear system of conservation laws of Temple class. In this paper, a continuous semigroup of solutions is constructed on a domain of ${{\bf L}}^\infty$ functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the ${{\bf L}}^1$ distance. Moreover, we show that a weak solution of the Cauchy problem coincides with the corresponding semigroup trajectory if and only if it satisfies an entropy condition of Oleinik type, concerning the decay of positive waves.

We consider the Cauchy problem for a nonlinear $n \times n$ system of conservation laws of Temple class, i.e., with coinciding shock and rarefaction curves and with a coordinate system made of Riemann invariants. Without any assumption on the convexity of the flux function, we prove the existence of a semigroup made of weak solutions of the equations and depending Lipschitz continuously on the initial data with bounded total variation.

Invariance properties of a scalar, linear heat equation with nonlocal boundary conditions are discussed as a function of a real parameter appearing in the boundary conditions of the problem. The equation is a model for a thermostat with sensor and controller positioned at opposite ends of an interval, whence the non-locality. It is shown that the analytic semigroup associated with the evolution problem is positive if and only if the parameter is in $(-\infty,0]\,$. For the corresponding elliptic problem three maximum principles are proved which hold for different parameter ranges.

We define the expressions $J\log^{1+\alpha}(e+|D f|)$ and $J\log^{1+\alpha}(e+|J|)$ as Schwartz distributions, for $f\colon\Omega\subset{{\Bbb R}}^n\to {{\Bbb R}}^n$ a Sobolev mapping such that $|D f|^n\log^\alpha(e+|D f|)$ is locally integrable, $-1 <\alpha <0$, and $J$ the Jacobian determinant.

The phenomenon of surface diffusion is of interest in a variety of physical situations [8]. Surface diffusion is modelled by a fourth-order quasilinear parabolic partial differential equation associated with the negative of the surface-Laplacian-of-curvature operator. We address the well-posedness of the corresponding initial value problem in the case in which the interface is a smooth closed curve $\Gamma$ contained in a tubular neighborhood of a fixed simple closed curve $\Gamma_0$ in the plane. We prove existence and uniqueness, as well as analytic dependence on the initial data of classical solutions of this problem locally in time, in the spaces $E^h$ of functions $f$ whose Fourier transform $(\hat f_k)_{k \in \bf Z}$ decays faster than $|k|^{-h}$, for $h > 5$. Our results are based on the machinery developed in [1], [2], and [3], which allows the application of the method of maximal regularity ([11], [14], and [4]) in the spaces $E^h$.