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Discrete & Continuous Dynamical Systems - S

May 2020 , Volume 13 , Issue 5

Issue on a tribute to Professor Angelo Favini on the occasion of his 70th birthday

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Preface to this special issue
Genni Fragnelli, Luca Lorenzi and Alain Miranville
2020, 13(5): ⅰ-ⅲ doi: 10.3934/dcdss.2020080 +[Abstract](710) +[HTML](132) +[PDF](1559.81KB)
Abstract:
An identification problem for a linear evolution equation in a banach space
Viorel Barbu and Gabriela Marinoschi
2020, 13(5): 1429-1440 doi: 10.3934/dcdss.2020081 +[Abstract](1414) +[HTML](520) +[PDF](320.68KB)
Abstract:

We study a problem of a parameter identification related to a linear evolution equation in a Banach space, using an additional information about the solution. For sufficiently regular data we provide an exact solution given by a Volterra integral equation, while for less regular data we obtain an approximating solution by an optimal control approach. Under certain hypotheses, the characterization of the limit of the sequence of the approximating solutions reveals that it is a solution to the original identification problem. An application to an inverse problem arising in population dynamics is presented.

Precise estimates for biorthogonal families under asymptotic gap conditions
Piermarco Cannarsa, Patrick Martinez and Judith Vancostenoble
2020, 13(5): 1441-1472 doi: 10.3934/dcdss.2020082 +[Abstract](1030) +[HTML](474) +[PDF](487.68KB)
Abstract:

A classical and useful way to study controllability problems is the moment method developed by Fattorini-Russell [12, 13], which is based on the construction of suitable biorthogonal families. Several recent problems exhibit the same behavior: the eigenvalues of the problem satisfy a uniform but rather 'bad' gap condition, and a rather 'good' but only asymptotic one. The goal of this work is to obtain general and precise upper and lower bounds for biorthogonal families under these two gap conditions, and thus to measure the influence of the 'bad' gap condition and the good influence of the 'good' asymptotic one. To achieve our goals, we extend some of the general results by Fattorini-Russell [12, 13] concerning biorthogonal families, using complex analysis techniques that were developed by Seidman [36], Güichal [20], Tenenbaum-Tucsnak [37] and Lissy [27, 28].

Global existence for Laplace reaction-diffusion equations
Angelo Favini and Atsushi Yagi
2020, 13(5): 1473-1493 doi: 10.3934/dcdss.2020083 +[Abstract](1474) +[HTML](455) +[PDF](362.04KB)
Abstract:

We study the initial-boundary value problem for a Laplace reaction-diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show global existence under suitable assumptions on the reaction function. We also show that the problem generates a dynamical system in a suitably set universal space and that this dynamical system possesses a Lyapunov function.

Singular parabolic equations with interior degeneracy and non smooth coefficients: The Neumann case
Genni Fragnelli and Dimitri Mugnai
2020, 13(5): 1495-1511 doi: 10.3934/dcdss.2020084 +[Abstract](1165) +[HTML](450) +[PDF](172.95KB)
Abstract:

We establish Hardy - Poincaré and Carleman estimates for non-smooth degenerate/singular parabolic operators in divergence form with Neumann boundary conditions. The degeneracy and the singularity occur both in the interior of the spatial domain. We apply these inequalities to deduce well-posedness and null controllability for the associated evolution problem.

A generalized Cox-Ingersoll-Ross equation with growing initial conditions
Giséle Ruiz Goldstein, Jerome A. Goldstein, Rosa Maria Mininni and Silvia Romanelli
2020, 13(5): 1513-1528 doi: 10.3934/dcdss.2020085 +[Abstract](984) +[HTML](419) +[PDF](410.26KB)
Abstract:

In this paper we solve the problem of the existence and strong continuity of the semigroup associated with the initial value problem for a generalized Cox-Ingersoll-Ross equation for the price of a zero-coupon bond (see [8]), on spaces of continuous functions on \begin{document}$ [0, \infty) $\end{document} which can grow at infinity. We focus on the Banach spaces

which contain the nonzero constants very common as initial data in the Cauchy problems coming from financial models. In addition, a Feynman-Kac type formula is given.

Vector-valued Schrödinger operators in Lp-spaces
Markus Kunze, Abdallah Maichine and Abdelaziz Rhandi
2020, 13(5): 1529-1541 doi: 10.3934/dcdss.2020086 +[Abstract](1152) +[HTML](405) +[PDF](342.41KB)
Abstract:

In this paper we consider the vector-valued operator div\begin{document}$ (Q\nabla u)-Vu $\end{document} of Schrödinger type. Here \begin{document}$ V = (v_{ij}) $\end{document} is a nonnegative, locally bounded, matrix-valued function and \begin{document}$ Q $\end{document} is a symmetric, strictly elliptic matrix whose entries are bounded and continuously differentiable with bounded derivatives. Concerning the potential \begin{document}$ V $\end{document}, we assume an that it is pointwise accretive and that its entries are in \begin{document}$ L^\infty_{{\rm loc}}( \mathbb{R}^d) $\end{document}. Under these assumptions, we prove that a realization of the vector-valued Schrödinger operator generates a \begin{document}$ C_0 $\end{document}-semigroup of contractions in \begin{document}$ L^p( \mathbb{R}^d; \mathbb{C}^m) $\end{document}. Further properties are also investigated.

Mean periodic solutions of a inhomogeneous heat equation with random coefficients
Galina Kurina and Vladimir Zadorozhniy
2020, 13(5): 1543-1551 doi: 10.3934/dcdss.2020087 +[Abstract](1139) +[HTML](450) +[PDF](255.63KB)
Abstract:

We present conditions ensuring the periodicity of the mathematical expectation of a solution of a scalar linear inhomogeneous heat equation with random coefficients where the coefficient in front of the unknown functions is Gaussian or it is uniformly distributed. The obtained results may be treated as finding a control ensuring the periodicity of the mathematical expectation of a solution of the heat equation.

The Stokes problem in fractal domains: Asymptotic behaviour of the solutions
Maria Rosaria Lancia and Paola Vernole
2020, 13(5): 1553-1565 doi: 10.3934/dcdss.2020088 +[Abstract](1092) +[HTML](416) +[PDF](680.45KB)
Abstract:

We study a Stokes problem in a three dimensional fractal domain of Koch type and in the corresponding prefractal approximating domains. We prove that the prefractal solutions do converge to the limit fractal one in a suitable sense. Namely the approximating velocity vector fields as well as the approximating associated pressures converge to the limit fractal ones respectively.

Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation
Costică Moroşanu
2020, 13(5): 1567-1587 doi: 10.3934/dcdss.2020089 +[Abstract](1282) +[HTML](408) +[PDF](427.07KB)
Abstract:

We present the error analysis of two time-stepping schemes of fractional steps type, used in the discretization of a nonlinear reaction-diffusion equation with Neumann boundary conditions, relevant in phase transition and interface problems. We start by investigating the solvability of a such boundary value problems in the class \begin{document}$ W^{1,2}_p(Q) $\end{document}. One proves the existence, the regularity and the uniqueness of solutions, in the presence of the cubic nonlinearity type. The convergence and error estimate results (using energy methods) for the iterative schemes of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual algorithm is formulated in the end. Numerical experiments are presented in order to validates the theoretical results (conditions of numerical stability) and to compare the accuracy of the methods.

Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory
Luciano Pandolfi
2020, 13(5): 1589-1599 doi: 10.3934/dcdss.2020090 +[Abstract](948) +[HTML](388) +[PDF](262.11KB)
Abstract:

In this paper we present a linear method for the identification of both the energy and flux relaxation kernels in the equation of thermodynamics with memory proposed by M.E. Gurtin and A.G. Pipkin. The method reduces the identification of the two kernels to the solution of two (linear) deconvolution problems. The energy relaxation kernel is reconstructed by means of energy measurements as the solution of a Volterra integral equation of the first kind which does not depend on the still unknown flux relaxation kernel. Then, flux measurements are used to identify the flux relaxation kernel.

The hypoelliptic Robin problem for quasilinear elliptic equations
Kazuaki Taira
2020, 13(5): 1601-1618 doi: 10.3934/dcdss.2020091 +[Abstract](1425) +[HTML](437) +[PDF](374.14KB)
Abstract:

This paper is devoted to the study of a hypoelliptic Robin boundary value problem for quasilinear, second-order elliptic differential equations depending nonlinearly on the gradient. More precisely, we prove an existence and uniqueness theorem for the quasilinear hypoelliptic Robin problem in the framework of Hölder spaces under the quadratic gradient growth condition on the nonlinear term. The proof is based on the comparison principle for quasilinear problems and the Leray–Schauder fixed point theorem. Our result extends earlier theorems due to Nagumo, Akô and Schmitt to the hypoelliptic Robin case which includes as particular cases the Dirichlet, Neumann and regular Robin problems.

Errata to "Modeling and optimal control of HIV/AIDS prevention through PrEP", Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 1,119–141
Cristiana J. Silva and Delfim F. M. Torres
2020, 13(5): 1619-1621 doi: 10.3934/dcdss.2020343 +[Abstract](443) +[HTML](71) +[PDF](203.19KB)
Abstract:

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