We prove a stability result for damped nonlinear wave equations,
when the damping changes sign and the nonlinear term satisfies a few
Inspired by a biological model on genetic repression
proposed by P. Jacob and J. Monod, we introduce a new class of delay
equations with nonautonomous past and nonlinear delay operator. With
the aid of some new techniques from functional analysis we prove
that these equations, which cover the biological model, are
We prove null controllability results for the one dimensional degenerate heat equation in non divergence form with a drift term and Neumann boundary conditions. To this aim we prove Carleman estimates for the associated adjoint problem.
extensions are considered.
This paper is devoted to study the well-posedness and the
asymptotic behavior of a population equation with diffusion in
$L^1$. The death and birth rates depend on the age and the spatial
variable. Here we allow the birth process to depend also on some
modified delay. This paper is a continuation of the studies done
by Nickel, Rhandi and Schnaubelt in  and
Fragnelli, Maniar, Piazzera and Tonetto in .
delay boundary conditions
We correct a flaw in the proof of [1, Lemma 2.3].
We give null controllability results for some degenerate parabolic equations in non divergence form with a drift term in one space dimension. In particular, the coefficient of the second order term may degenerate at the extreme points of the space domain. For this purpose, we obtain an observability inequality for the adjoint problem using suitable Carleman estimates.
We consider a nonlinear elliptic equation with Robin boundary condition driven by the p-Laplacian and with a reaction term which depends also on the gradient. By using a topological approach based on the Leray-Schauder alternative principle, we show the existence of a smooth solution.
The aim of the paper is to provide conditions ensuring the
existence of non-trivial non-negative periodic solutions to a
system of doubly degenerate parabolic equations containing delayed
nonlocal terms and satisfying Dirichlet boundary conditions. The
employed approach is based on the theory of the Leray-Schauder
topological degree theory, thus a crucial purpose of the paper is
to obtain a priori bounds in a convenient functional space, here
$L^2(Q_T)$, on the solutions of certain homotopies. This is
achieved under different assumptions on the sign of the kernels of
the nonlocal terms. The considered system is a possible model of
the interactions between two biological species sharing the same
territory where such interactions are modeled by the kernels of
the nonlocal terms. To this regard the obtained results can be
viewed as coexistence results of the two biological populations
under different intra and inter specific interferences on their
natural growth rates.