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DCDS

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 well--posed.
DCDS-S

Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates

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.
Some linear
extensions are considered.

DCDS-S

We prove a stability result for damped nonlinear wave equations,
when the damping changes sign and the nonlinear term satisfies a few
natural assumptions.

NHM

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.

DCDS

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.

DCDS-B

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 [28][32][33] and
Fragnelli, Maniar, Piazzera and Tonetto in [15][21][29][30].

DCDS

We correct a flaw in the proof of [1, Lemma 2.3].

DCDS-S

We consider operators in divergence form, $A_1u=(au')'$, and in nondivergence form, $A_2u=au''$, provided that the coefficient $a$ vanishes in an interior point of the space domain. Characterizing the domain of the operators, we prove that, under suitable assumptions, the operators $A_1$ and $A_2$, equipped with general Wentzell boundary conditions, are nonpositive and selfadjoint on spaces of $L^2$ type.

DCDS-S

In the study of mathematical models, which lead to
Cauchy problems for differential equations of parabolic (resp. hyperbolic) type
or to an elliptic boundary value problem the following issues typically have
a prominent interest:

For more information please click the “Full Text” above.

For more information please click the “Full Text” above.

keywords:

DCDS

We consider a parametric nonlinear Robin problem driven by the $p -$Laplacian plus an indefinite potential and a Carathéodory reaction which is $(p-1) -$ superlinear without satisfying the Ambrosetti - Rabinowitz condition. We prove a bifurcation-type result describing the dependence of the set of positive solutions on the parameter. We also prove the existence of nodal solutions. Our proofs use tools from critical point theory, Morse theory and suitable truncation techniques.

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