On a nonlocal degenerate parabolic problem

Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved

Generalized Kelvin-Voigt equations for nonhomogeneous andincompressible fluids

In this work, we consider the Kelvin-Voigt equations for non-homogeneous and incompressible fluid flows with the diffusion and relaxation terms described by two distinct power-laws. Moreover, we assume that the momentum equation is perturbed by an extra term, which, depending on whether its signal is positive or negative, may account for the presence of a source or a sink within the system. For the associated initial-boundary value problem, we study the existence of weak solutions as well as the large-time behavior of the solutions. In the case the extra term is a sink, we prove the global existence of weak solutions and we establish the conditions for the polynomial time decay and for the exponential time decay of these solutions. If the extra term is a source, we show how the exponents of nonlinearity must interact to ensure the local existence of weak solutions.Russian Federation government [14.W03.31.0002]Portuguese Foundation for Science and Technology (FCT), PortugalPortuguese Foundation for Science and Technology [UID/MAT/04561/2019]Portuguese Foundation for Science and Technology (FCT)Portuguese Foundation for Science and Technology [SFRH/BSAB/135242/2017]Ministry of Science and Education of the Republic of Kazakhstan (MES RK), Kazakhstan [AP08052425

New unilateral problems in stratigraphy

This work deals with the study of some stratigraphic models for the formation of geological basins under a maximal erosion rate constrain. It leads to introduce differential inclusions of degenerated hyperbolic-parabolic type $0\in \partial _{t}u-div\{H(\partial _{t}u+E)\nabla u\}$, where H is the maximal monotonous graph of the Heaviside function and E is a given non-negative function. Firstly, we present the new and realistic models and an original mathematical formulation, taking into account the weather-limited rate constraint in the conservation law, with a unilateral constraint on the outflow boundary. Then, we give a study of the 1-D case with numerical illustrations

Discrete solutions for the porous medium equation with absorption and variable exponents

In this work, we study the convergence of the finite element method when applied to the following parabolic equation: Since the equation may be of degenerate type, we use an approximate problem, regularized by introducing a parameter ε. We prove, under certain conditions on γ, σ and f, that the weak solution of the approximate problem converges to the weak solution of the initial problem, when the parameter ε tends to zero. The convergence of the discrete solutions for the weak solution of the approximate problem is also proved. Finally, we present some numerical results of a MatLab implementation of the method.No¯ 15-11-20019–financed by the Russian Science Foundation, Russiainfo:eu-repo/semantics/publishedVersio

On the finite element method for a nonlocal degenerate parabolic problem

The aim of this paper is the numerical study of a class of nonlinear nonlocal degenerate parabolic equations. The convergence and error bounds of the solutions are proved for a linearized Crank–Nicolson–Galerkin finite element method with polynomial approximations of degree k≥1. Some explicit solutions are obtained and used to test the implementation of the method in Matlab environment.15-11-20019 - financed by the Russian Science Fundation, Russiainfo:eu-repo/semantics/publishedVersio

Hardy Spaces with Variable Exponents

In this paper, we gather the essential contents of the talk ?Hardy spaces with variable exponents given by the second author at the CIMPA2017 Research School-IXEscuela Santaló Harmonic Analysis, Geometric Measure Theory and Applications,which took place in Buenos Aires, Argentina, from July 31 to August 11, 2017. Wegive an overview of the latest advances about Hardy and local Hardy spaces withvariable exponents in different contexts. We cannot be exhaustive and we recommendthe interested reader to consult the references at the end of this paper and others that can be found in them.Fil: Almeida, Víctor. Universidad de La Laguna; EspañaFil: Betancor, Jorge. Universidad de La Laguna; EspañaFil: Dalmasso, Estefanía Dafne. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Rodríguez Mesa, Lourdes. Universidad de La Laguna; Españ