DCDS
Nonlinear delay equations with nonautonomous past
Genni Fragnelli Dimitri Mugnai
Discrete & Continuous Dynamical Systems - A 2008, 21(4): 1159-1183 doi: 10.3934/dcds.2008.21.1159
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.
keywords: evolution family nonlinear delay equations genetic repression. local semigroup
DCDS-S
Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates
Genni Fragnelli
Discrete & Continuous Dynamical Systems - S 2013, 6(3): 687-701 doi: 10.3934/dcdss.2013.6.687
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.
keywords: observability Carleman estimates Hardy type inequality. Degenerate parabolic equations null controllability
DCDS-S
Stability of solutions for nonlinear wave equations with a positive--negative damping
Genni Fragnelli Dimitri Mugnai
Discrete & Continuous Dynamical Systems - S 2011, 4(3): 615-622 doi: 10.3934/dcdss.2011.4.615
We prove a stability result for damped nonlinear wave equations, when the damping changes sign and the nonlinear term satisfies a few natural assumptions.
keywords: positive-negative damping. Damped nonlinear wave equations
NHM
Null controllability of degenerate parabolic operators with drift
Piermarco Cannarsa Genni Fragnelli Dario Rocchetti
Networks & Heterogeneous Media 2007, 2(4): 695-715 doi: 10.3934/nhm.2007.2.695
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.
keywords: Carleman estimates Degenerate parabolic equations null controllability Hardy type inequality
DCDS
Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms
Genni Fragnelli Paolo Nistri Duccio Papini
Discrete & Continuous Dynamical Systems - A 2011, 31(1): 35-64 doi: 10.3934/dcds.2011.31.35
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.
keywords: topological degree. Doubly degenerate parabolic equations non-negative periodic solutions
DCDS-B
The asymptotic behavior of a population equation with diffusion and delayed birth process
Genni Fragnelli A. Idrissi L. Maniar
Discrete & Continuous Dynamical Systems - B 2007, 7(4): 735-754 doi: 10.3934/dcdsb.2007.7.735
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].
keywords: extrapolation theory delay boundary conditions perturbation evolution family backward asymptotic behavior. essential spectrum Population equation Dyson-Phillips expansion evolution semigroup semigroup
DCDS
Corrigendum: Nnon-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms
Genni Fragnelli Paolo Nistri Duccio Papini
Discrete & Continuous Dynamical Systems - A 2013, 33(8): 3831-3834 doi: 10.3934/dcds.2013.33.3831
We correct a flaw in the proof of [1, Lemma 2.3].
keywords: topological degree. non-negative periodic solutions Doubly degenerate parabolic equations
DCDS-S
Robin problems for the p-Laplacian with gradient dependence
Genni Fragnelli Dimitri Mugnai Nikolaos S. Papageorgiou
Discrete & Continuous Dynamical Systems - S 2019, 12(2): 287-295 doi: 10.3934/dcdss.2019020

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.

keywords: Convection term nonlinear regularity Leray-Schauder alternative principle maximal monotone map compact embedding
DCDS-S
Generalized Wentzell boundary conditions for second order operators with interior degeneracy
Genni Fragnelli Gisèle Ruiz Goldstein Jerome Goldstein Rosa Maria Mininni Silvia Romanelli
Discrete & Continuous Dynamical Systems - S 2016, 9(3): 697-715 doi: 10.3934/dcdss.2016023
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.
keywords: Second order operators in divergence and nondivergence form interior degeneracy generalized Wentzell boundary conditions.
DCDS-S
Preface to the special issue in memory of Alfredo Lorenzi
Angelo Favini Genni Fragnelli Luca Lorenzi
Discrete & Continuous Dynamical Systems - S 2016, 9(3): i-ii doi: 10.3934/dcdss.201603i
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:

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