ISSN:

1937-1632

eISSN:

1937-1179

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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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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.

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A classical and useful way to study controllability problems is the moment method developed by Fattorini-Russell [

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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.

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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.

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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 [

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.

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In this paper we consider the vector-valued operator div

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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.

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**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.

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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

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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.

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**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.

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