scholarly journals A non-smooth regularization of a forward–backward parabolic equation

2017 ◽  
Vol 27 (04) ◽  
pp. 641-661 ◽  
Author(s):  
Elena Bonetti ◽  
Pierluigi Colli ◽  
Giuseppe Tomassetti
Keyword(s):  
Parabolic Equation ◽  
Chemical Potential ◽  
Time Derivative ◽  
Maximal Monotone ◽  
Free Energy Density ◽  
Uniqueness Of Solutions ◽  
Maximal Monotone Graph ◽  
Viscous Regularization ◽  
Physical Quantities ◽  
Monotone Graph

In this paper, we introduce a model describing diffusion of species by a suitable regularization of a “forward–backward” parabolic equation. In particular, we prove existence and uniqueness of solutions, as well as continuous dependence on data, for a system of partial differential equations and inclusion, which may be interpreted, e.g. as evolving equation for physical quantities such as concentration and chemical potential. The model deals with a constant mobility and it is recovered from a possibly non-convex free-energy density. In particular, we render a general viscous regularization via a maximal monotone graph acting on the time derivative of the concentration and presenting a strong coerciveness property.

scholarly journals On the Existence of Solutions of a Two-Layer Green Roof Mathematical Model

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1608
Author(s):  
J. Ignacio Tello ◽  
Lourdes Tello ◽  
María Luisa Vilar
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence Of Solutions ◽  
Green Roof ◽  
Maximal Monotone ◽  
Equation Modeling ◽  
Partial Differential ◽  
Maximal Monotone Graph ◽  
Monotone Graph

The aim of this article is to fill part of the existing gap between the mathematical modeling of a green roof and its computational treatment, focusing on the mathematical analysis. We first introduce a two-dimensional mathematical model of the thermal behavior of an extensive green roof based on previous models and secondly we analyze such a system of partial differential equations. The model is based on an energy balance for buildings with vegetation cover and it is presented for general shapes of roofs. The model considers a vegetable layer and the substratum and the energy exchange between them. The unknowns of the problem are the temperature of each layer described by a coupled system of two partial differential equations of parabolic type. The equation modeling the evolution of the temperature of the substratum also considers the change of phase of water described by a maximal monotone graph. The main result of the article is the proof of the existence of solutions of the system which is given in detail by using a regularization of the maximal monotone graph. Appropriate estimates are obtained to pass to the limit in a weak formulation of the problem. The result goes one step further from modeling to validate future numerical results.

Almost-periodic forcing for a wave equation with a nonlinear, local damping term

1983 ◽  
Vol 94 (3-4) ◽  
pp. 195-212 ◽  
Author(s):  
Alain Haraux
Keyword(s):  
Maximal Monotone ◽  
Nonlinear Hyperbolic Equation ◽  
List Type ◽  
Almost Periodic ◽  
Damping Term ◽  
Periodic Forcing ◽  
Maximal Monotone Graph ◽  
Image Position ◽  
Uniformly Continuous ◽  
Monotone Graph

SynopsisLet Ω⊂ℝnbe a bounded open domain and T = ∂Ω. It β is a maximal monotone graph in ℝ×ℝ with 0ϵβ(0), and f: ℝ×Ω→ℝ is measurable with t→ f(t,.) S2-almost periodic as a function ℝ→L2(Ω), we consider the nonlinear hyperbolic equationWe show that:(i) if ゲ is strictly increasing and (1) has a solution ω on ℝ with [ω, Əω/Ət] almost periodic: , for any solution of (1) there exists with u(t,.)–ω(t,.)—ξin (ii) if β is single valued and everywhere defined, the existence of ω as above implies that, for every solution of (1), there exists Ϛ(t, x) with ә2Ϛ/әt2–0△Ϛ = in ℝ×Ω and u(t,.)–ω(t,.)—0 in as t → +∞(iii) if β–1 is uniformly continuous and ゲ satisfies some growth assumption (depending on N), for every f as above, there exists ω solution of (1) on ℝ with [ω, Əω/Ət] almost periodic: ℝ → .

scholarly journals Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition

2017 ◽  
Vol 8 (1) ◽  
pp. 679-693 ◽  
Author(s):  
Jesus Ildefonso Díaz ◽  
David Gómez-Castro ◽  
Alexander V. Podol’skii ◽  
Tatiana A. Shaposhnikova
Keyword(s):  
Boundary Condition ◽  
Chemical Engineering ◽  
Dirichlet Boundary ◽  
Critical Size ◽  
Maximal Monotone ◽  
Dirichlet Boundary Conditions ◽  
Diffusion Operator ◽  
Maximal Monotone Graph ◽  
Monotone Graph ◽  
Laplace Diffusion

Abstract The aim of this paper is to consider the asymptotic behavior of boundary value problems in n-dimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a p-Laplace diffusion operator on the interior, in the case in which the particles are of critical size. We consider the cases in which {1<p<n} , {n\geq 3} . In fact, in contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately. We characterize the so called “strange term” in the homogenized problem for the case in which the particles are balls of critical size. Moreover, by studying an application in Chemical Engineering, we show that the critically sized particles lead to a more effective homogeneous reaction than noncritically sized particles.

Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities

1991 ◽  
Vol 119 (1-2) ◽  
pp. 1-17 ◽  
Author(s):  
Eduard Feireisl ◽  
John Norbury
Keyword(s):  
Initial Data ◽  
Parabolic Equations ◽  
Initial Function ◽  
Maximal Monotone ◽  
Step Function ◽  
Heaviside Step Function ◽  
Discontinuous Nonlinearities ◽  
Maximal Monotone Graph ◽  
Image Position ◽  
Monotone Graph

SynopsisWe consider the problemwhere H stands for the maximal monotone graph associated with the Heaviside step function. It is shown that the problem possesses at least one (strong) solution belonging to an appropriate function space. Moreover, we prove:(i) There is a smooth initial function u0, u0≧1, where the equality holds at exactly one point such that there are at least two different solutions corresponding to the initial data u0.(ii) The comparison principle: The relation for any x ≠ 0, implies u1(t)>u2(t), t≧0 for any u1, u2 solving the problem with the initial data , respectively.(iii) For a “reasonable” set of initial data the solution is uniquely determined. Moreover, the free boundary {(x, t)| u(x, t) = 1} is regular and on its complement the equation holds in a classical sense.

scholarly journals Nonlinear elliptic anisotropic problem involving non-local boundary conditions with variable exponent and graph data

2020 ◽  
Vol 29 (2) ◽  
pp. 145-152
Author(s):  
ADAMA KABORE ◽  
STANISLAS OUARO
Keyword(s):  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Maximal Monotone ◽  
Local Conditions ◽  
Graph Data ◽  
Local Boundary ◽  
Nonlinear Elliptic ◽  
Maximal Monotone Graph ◽  
Non Local ◽  
Monotone Graph

We study a nonlinear elliptic anisotropic problem involving non-local conditions. We also consider variable exponent and general maximal monotone graph datum at the boundary. We prove the existence and uniqueness of weak solution to the problem.

Multivalued Stochastic Differential Equations and Its Applications in Dynamics

2003 ◽  
Author(s):  
F. Bernardin ◽  
C. H. Lamarque ◽  
M. Schatzman
Keyword(s):  
Dry Friction ◽  
Existence And Uniqueness ◽  
Numerical Scheme ◽  
Maximal Monotone ◽  
Nonlinear Dynamical ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Maximal Monotone Graph ◽  
Stochastic Term ◽  
Monotone Graph

Using a maximal monotone graph we can deal with many nonlinear dynamical problems involving e.g. dry friction or impacts. When submitted to an external stochastic term, a class of differential inclusions is obtained. Existence and uniqueness results are recalled. An adapted numerical scheme is presented. Numerical results are presented for different examples of models.

Solution of the nonregular problem for a parabolic equation with the time derivative in the boundary condition

2021 ◽  
pp. 1-35
Author(s):  
Galina Bizhanova
Keyword(s):  
Boundary Layer ◽  
Boundary Condition ◽  
Parabolic Equation ◽  
Small Parameter ◽  
Unique Solvability ◽  
Time Derivative ◽  
Holder Space ◽  
Hölder Space ◽  
Space Solution ◽  
Unperturbed Problem

There is studied the Hölder space solution u ε of the problem for parabolic equation with the time derivative ε ∂ t u ε | Σ in the boundary condition, where ε > 0 is a small parameter. The unique solvability of the perturbed problem and estimates of it’s solution are obtained. The convergence of u ε as ε → 0 to the solution of the unperturbed problem is proved. Boundary layer is not appeared.

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