It is shown that every result about a local degree and thus about bifurcation for variational inequalities (which is usually an obstacle problem) leads to a corresponding result for a problem in which the obstacle is modeled by inclusions instead of inequalities. Using recent results about variational inequalities, we obtain in particular bifurcation of stationary spatially nonhomogeneous solutions of a reaction-diffusion system with only Neumann conditions and obstacles modeled by inclusions, and results for an elliptic equation with inclusions about bifurcation strictly between certain eigenvalues and also bifurcation at eigenvalues without multiplicity assumptions.

In this paper we prove the existence of concentrating solutions for a slightly sub-critical problem involving the Kohn Laplacian on a bounded domain of the Heisenberg group, under the assumption that the Robin's function of the domain has a non-degenerate critical point.

We study the initial-value problem for the nonlinear Schrödinger equation $$ i\partial _{t}u+\Delta u=\lambda\vert u\vert ^{p}, \quad\left( t,x\right) \in \left[ 0,T\right) \times \mathbf{R}^{n}, $$ where $1 < p$ and $\lambda\in\mathbf{C}\setminus\{0\}$. The local well-posedness is well known in $L^2$ if $1 < p < 1+4/n$. In this paper, we study the global behavior of the solutions, and we will prove a small-data blow-up result of an $L^2$-solution when $1 < p\le 1+2/n$.

We consider a large class of retarded functional differential equations subject to impulse effects at variable times, and we present an averaging result for this class of equations by means of the techniques and tools of the theory of generalized ordinary differential equations introduced by J. Kurzweil.

In this work, we will be interested in the study of chaotic behavior exhibited by some linear infinite-dimensional systems. Our tool for this is the theory of chaotic linear semigroups of operators, and we will make use of the imaginary eigenvalues of the infinitesimal generator to construct an invariant Gaussian measure with respect to which the associated semigroup will be strong mixing. An application to the dynamic of a size-structured cell population is given.

We study a nonlocal version of the two-phase Stefan problem, which models a phase-transition problem between two distinct phases evolving to distinct heat equations. Mathematically speaking, this consists in deriving a theory for sign-changing solutions of the equation, $u_t=J\ast v -v $, $v=\Gamma(u)$, where the monotone graph is given by $\Gamma(s)=\rm{sign}(s)(|s|-1)_+$. We give general results of existence, uniqueness and comparison, in the spirit of [2]. Then we focus on the study of the asymptotic behavior for sign-changing solutions, which present challenging difficulties due to the nonmonotone evolution of each phase.

This paper deals with the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem: \begin{equation*} \left( -\Delta _{\mid D}\right) ^{\frac{\alpha }{2}}u=a_{1}(x)u^{\sigma _{1}}+a_{2}(x)u^{\sigma _{2}}\text{ in }D,\text{ }\underset{x\rightarrow z\in \partial D}{\lim }\left( \delta (x)\right) ^{2-\alpha }u(x)=0, \end{equation*} where $0 < \alpha < 2,$ $\sigma _{1},\sigma _{2}\in (-1,1),$ $D$ is a bounded $C^{1,1}$-domain in $\mathbb{R}^{n},$ $n\geq 2,$ and $\delta (x)$ denotes the Euclidian distance from $x$ to the boundary of $D$. The nonnegative weight functions $a_{1}$ and $a_{2}$ are in $C_{loc}^{\gamma }(D),$ $ 0 < \gamma < 1,$ satisfying some appropriate assumptions related to Karamata regular variation theory. We also give the global behavior of such a solution.

The Aleksandrov--Bakelman--Pucci maximum principle is established for $L^p$-viscosity solutions of fully nonlinear second-order elliptic PDEs having linear and superlinear growth terms for the first derivatives with small coefficients.

In this paper, we are interested in the null controllability of a linear population dynamics model with degenerate dispersion coefficient. We develop first a Carleman-type inequality for its adjoint system, and then an observability inequality which allows us to establish the existence of a control acting on a subset of the space domain which steers the population of a certain age to extinction in a finite time.

We prove the existence of heteroclinic solutions of the prescribed curva\-ture equation \begin{equation*} \Big( u'/{ \sqrt{1+{u'}^2}}\Big)' = a(t)V'(u), \end{equation*} where $V$ is a double-well potential and $a$ is asymptotic to a positive periodic function. Such an equation is meaningful in the modeling theory of reaction-diffusion phenomena which feature saturation at large value of the gradient. According to numerical simulations (see [5]), the graph of the interface between the stable states of a two-phase system may exhibit discontinuities. We provide a theoretical justification of these simulations by showing that an optimal transition between the stable states arises as a minimum of the associated action functional in the space of locally bounded variation functions. In very simple cases, such an optimal transition naturally displays jumps.