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Discrete & Continuous Dynamical Systems - S

2016 , Volume 9 , Issue 3

Issue in memory of Alfredo Lorenzi

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Preface to the special issue in memory of Alfredo Lorenzi
Angelo Favini, Genni Fragnelli and Luca Lorenzi
2016, 9(3): i-ii doi: 10.3934/dcdss.201603i +[Abstract](405) +[PDF](91.4KB)
Abstract:
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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On some boundary control problems
Orazio Arena
2016, 9(3): 613-618 doi: 10.3934/dcdss.2016015 +[Abstract](403) +[PDF](286.6KB)
Abstract:
The boundary controllability problems firstly discussed, in this paper, might be described by a one-dimensional $x$-space equation and $t>0$, modeling - at same time $t$ - different physical phenomena in a composite solid made of different materials. These phenomena may be governed, at same time $t$, for example, by the heat equation and by the Schrödinger equation in separate regions. Interface conditions are assumed. Extensions of such boundary controllability problems to two-dimensional $(x,y)$-space are also investigated.
A fractional eigenvalue problem in $\mathbb{R}^N$
Giacomo Bocerani and Dimitri Mugnai
2016, 9(3): 619-629 doi: 10.3934/dcdss.2016016 +[Abstract](403) +[PDF](388.7KB)
Abstract:
We prove that a linear fractional operator with an asymptotically constant lower order term in the whole space admits eigenvalues.
Periodic solutions to nonlocal MEMS equations
Daniele Cassani and Antonio Tarsia
2016, 9(3): 631-642 doi: 10.3934/dcdss.2016017 +[Abstract](427) +[PDF](394.9KB)
Abstract:
Combining a priori estimates with penalization techniques and an implicit function argument based on Campanato's near operators theory, we obtain the existence of periodic solutions for a fourth order integro-differential equation modelling actuators in MEMS devices.
The problem of detecting corrosion by an electric measurement revisited
Mourad Choulli and Aymen Jbalia
2016, 9(3): 643-650 doi: 10.3934/dcdss.2016018 +[Abstract](384) +[PDF](359.0KB)
Abstract:
We establish a logarithmic stability estimate for the problem of detecting corrosion by a single electric measurement. We give a proof based on an adaptation of the method initiated in [3] for solving the inverse problem of recovering the surface impedance of an obstacle from the scattering amplitude. The key idea consists in estimating accurately a lower bound of the local $L^2$-norm at the boundary, of the solution of the boundary value problem used in modeling the problem of detection corrosion by an electric measurement.
Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations
Raluca Clendenen, Gisèle Ruiz Goldstein and Jerome A. Goldstein
2016, 9(3): 651-660 doi: 10.3934/dcdss.2016019 +[Abstract](331) +[PDF](328.8KB)
Abstract:
In the dynamic or Wentzell boundary condition for elliptic, parabolic and hyperbolic partial differential equations, the positive flux coefficient $% \beta $ determines the weighted surface measure $dS/\beta $ on the boundary of the given spatial domain, in the appropriate Hilbert space that makes the generator for the problem selfadjoint. Usually, $\beta $ is continuous and bounded away from both zero and infinity, and thus $L^{2}\left( \partial \Omega ,dS\right) $ and $L^{2}\left( \partial \Omega ,dS/\beta \right) $ are equal as sets. In this paper this restriction is eliminated, so that both zero and infinity are allowed to be limiting values for $\beta $. An application includes the parabolic asymptotics for the Wentzell telegraph equation and strongly damped Wentzell wave equation with general $\beta $.
A singular limit problem for the Ibragimov-Shabat equation
Giuseppe Maria Coclite and Lorenzo di Ruvo
2016, 9(3): 661-673 doi: 10.3934/dcdss.2016020 +[Abstract](367) +[PDF](374.4KB)
Abstract:
We consider the Ibragimov-Shabat equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.
$1$-dimensional Harnack estimates
Fatma Gamze Düzgün, Ugo Gianazza and Vincenzo Vespri
2016, 9(3): 675-685 doi: 10.3934/dcdss.2016021 +[Abstract](386) +[PDF](386.9KB)
Abstract:
Let $u$ be a non-negative super-solution to a $1$-dimensional singular parabolic equation of $p$-Laplacian type ($1< p <2$). If $u$ is bounded below on a time-segment $\{y\}\times(0,T]$ by a positive number $M$, then it has a power-like decay of order $\frac p{2-p}$ with respect to the space variable $x$ in $\mathbb R\times[T/2,T]$. This fact, stated quantitatively in Proposition 1.2, is a ``sidewise spreading of positivity'' of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an effect is based on geometrical ideas.
Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel
Vladimir E. Fedorov and Natalia D. Ivanova
2016, 9(3): 687-696 doi: 10.3934/dcdss.2016022 +[Abstract](443) +[PDF](372.2KB)
Abstract:
An identification problem is considered for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Solutions of problems with Cauchy and Showalter conditions on initial values are proved to be existing and unique. Solutions stability estimates are derived. The abstract results are applied to an identification problem for the linearized Oskolkov system of equations. There are considered different degrees of system degeneration with respect to the time derivatives of unknown functions.
Generalized Wentzell boundary conditions for second order operators with interior degeneracy
Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni and Silvia Romanelli
2016, 9(3): 697-715 doi: 10.3934/dcdss.2016023 +[Abstract](379) +[PDF](417.6KB)
Abstract:
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.
Classical solutions to quasilinear parabolic problems with dynamic boundary conditions
Davide Guidetti
2016, 9(3): 717-736 doi: 10.3934/dcdss.2016024 +[Abstract](372) +[PDF](389.0KB)
Abstract:
We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear dynamic boundary conditions.
Inverse problems for evolution equations with time dependent operator-coefficients
Mohammed Al Horani, Angelo Favini and Hiroki Tanabe
2016, 9(3): 737-744 doi: 10.3934/dcdss.2016025 +[Abstract](347) +[PDF](308.5KB)
Abstract:
In this paper we study an inverse problem with time dependent operator-coefficients. We indicate sufficient conditions for the existence and the uniqueness of a solution to this problem. A number of concrete applications to partial differential equations is described.
Observability of $N$-dimensional integro-differential systems
Paola Loreti and Daniela Sforza
2016, 9(3): 745-757 doi: 10.3934/dcdss.2016026 +[Abstract](449) +[PDF](371.6KB)
Abstract:
The aim of the paper is to show a reachability result for the solution of a multidimensional coupled Petrovsky and wave system when a non local term, expressed as a convolution integral, is active. Motivations to the study are in linear acoustic theory in three dimensions. To achieve that, we prove observability estimates by means of Ingham type inequalities applied to the Fourier series expansion of the solution.
Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions
Ahmad Makki and Alain Miranville
2016, 9(3): 759-775 doi: 10.3934/dcdss.2016027 +[Abstract](454) +[PDF](382.5KB)
Abstract:
Our aim in this paper is to prove the existence and uniqueness of solutions to Cahn-Hilliard and Allen-Cahn type equations based on a modification of the Ginzburg-Landau free energy proposed in [12] (see also [16]) which takes into account strong anisotropy effects. In particular, the free energy contains a regularization term, called Willmore regularization.
Semigroup-theoretic approach to identification of linear diffusion coefficients
Gianluca Mola, Noboru Okazawa, Jan Prüss and Tomomi Yokota
2016, 9(3): 777-790 doi: 10.3934/dcdss.2016028 +[Abstract](388) +[PDF](406.8KB)
Abstract:
Let $X$ be a complex Banach space and $A:\,D(A) \to X$ a quasi-$m$-sectorial operator in $X$. This paper is concerned with the identification of diffusion coefficients $\nu > 0$ in the initial-value problem: \[ (d/dt)u(t) + {\nu}Au(t) = 0, \quad t \in (0,T), \quad u(0) = x \in X, \] with additional condition $\|u(T)\| = \rho$, where $\rho >0$ is known. Except for the additional condition, the solution to the initial-value problem is given by $u(t) := e^{-t\,{\nu}A} x \in C([0,T];X) \cap C^{1}((0,T];X)$. Therefore, the identification of $\nu$ is reduced to solving the equation $\|e^{-{\nu}TA}x\| = \rho$. It will be shown that the unique root $\nu = \nu(x,\rho)$ depends on $(x,\rho)$ locally Lipschitz continuously if the datum $(x,\rho)$ fulfills the restriction $\|x\|> \rho$. This extends those results in Mola [6](2011).
Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback
Serge Nicaise and Cristina Pignotti
2016, 9(3): 791-813 doi: 10.3934/dcdss.2016029 +[Abstract](479) +[PDF](433.7KB)
Abstract:
We study the stabilization problem for the wave equation with localized Kelvin--Voigt damping and mixed boundary condition with time delay. By using a frequency domain approach we show that, under an appropriate condition between the internal damping and the boundary feedback, an exponential stability result holds. In this sense, this extends the result of [19] where, in a more general setting, the case of distributed structural damping is considered.
A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications
Matteo Novaga, Diego Pallara and Yannick Sire
2016, 9(3): 815-831 doi: 10.3934/dcdss.2016030 +[Abstract](417) +[PDF](440.4KB)
Abstract:
The purpose of this paper is to study a boundary reaction problem on the space $X \times {\mathbb R}$, where $X$ is an abstract Wiener space. We prove that smooth bounded solutions enjoy a symmetry property, i.e., are one-dimensional in a suitable sense. As a corollary of our result, we obtain a symmetry property for some solutions of the following equation $$ (-\Delta_\gamma)^s u= f(u), $$ with $s\in (0,1)$, where $(-\Delta_\gamma)^s$ denotes a fractional power of the Ornstein-Uhlenbeck operator, and we prove that for any $s \in (0,1)$ monotone solutions are one-dimensional.
Strong solutions of quasilinear equations in Banach spaces not solvable with respect to the highest-order derivative
Marina V. Plekhanova
2016, 9(3): 833-846 doi: 10.3934/dcdss.2016031 +[Abstract](355) +[PDF](382.6KB)
Abstract:
By means of the Mittag-Leffler function existence and uniqueness conditions are obtained for a strong solution of the Cauchy problem to quasilinear differential equation in a Banach space, solved with respect to the highest-order derivative. The results are used in the study of quasilinear equations with degenerate operator at the highest-order derivative. Some special restrictions for nonlinear operator in the equation are used here. Existence conditions of a unique strong solution for the Cauchy problem and generalized Showalter--Sidorov for degenerate quasilinear equations were found. The obtained results are illustrated by an example of initial-boundary value problem for a quasilinear system of equations not resolved with respect to the highest-order time derivative.
Nonlocal elliptic problems in infinite cylinder and applications
Alexander L. Skubachevskii
2016, 9(3): 847-868 doi: 10.3934/dcdss.2016032 +[Abstract](363) +[PDF](497.7KB)
Abstract:
We consider a unique solvability of nonlocal elliptic problems in infinite cylinder in weighted spaces and in Hölder spaces. Using these results we prove the existence and uniqueness of classical solution for the Vlasov--Poisson equations with nonlocal conditions in infinite cylinder for sufficiently small initial data.
On nonlinear and quasiliniear elliptic functional differential equations
Olesya V. Solonukha
2016, 9(3): 869-893 doi: 10.3934/dcdss.2016033 +[Abstract](361) +[PDF](433.9KB)
Abstract:
We consider nonlinear elliptic functional differential equations. The corresponding operator has the form of a product of nonlinear elliptic differential mapping and linear difference mapping. It were obtained sufficient conditions for solvability of the Dirichlet problem. A concrete example shows that a nonlinear differential--difference operator may not be strongly elliptic even if the nonlinear differential operator is strongly elliptic and the linear difference operator is positive definite. The analysis is based on the theory of pseudomonotone--type operators and linear theory of elliptic functional differential operators.

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