Multiscale homogenization of nonlinear hyperbolic-parabolic equations

Abdelhakim Dehamnia; Hamid Haddadou

Displaying similar documents to “Multiscale homogenization of nonlinear hyperbolic-parabolic equations”

Homogenization of a linear parabolic problem with a certain type of matching between the microscopic scales

Pernilla Johnsen, Tatiana Lobkova (2018)

Applications of Mathematics

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This paper is devoted to the study of the linear parabolic problem $ε \partial_{t} u_{ε} (x, t) - \nabla \cdot (a (x / ε, t / ε^{3}) \nabla u_{ε} (x, t)) = f (x, t)$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient $ε$ in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic...

$L^{p} - L^{q}$ time decay estimates for the solution of the linear partial differential equations of thermodiffusion

Arkadiusz Szymaniec (2010)

Applicationes Mathematicae

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We consider the initial-value problem for a linear hyperbolic parabolic system of three coupled partial differential equations of second order describing the process of thermodiffusion in a solid body (in one-dimensional space). We prove $L^{p} - L^{q}$ time decay estimates for the solution of the associated linear Cauchy problem.

Global solutions to initial value problems in nonlinear hyperbolic thermoelasticity

Jerzy August Gawinecki

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CONTENTS1. Introduction..................................................................................................................................... 5 1.1. Main Theorem 1.1................................................................................................................. 8 1.2. Main Theorem 1.2................................................................................................................. 92. Radon transform.......................................................................................................................................

Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations

Peter Poláčik (2002)

Mathematica Bohemica

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We consider three types of semilinear second order PDEs on a cylindrical domain $Ω \times (0, \infty)$ , where $Ω$ is a bounded domain in $ℝ^{N}$ , $N \geq 2$ . Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of $Ω \times (0, \infty)$ is reserved for time $t$ , the third type is an elliptic equation with a singled out unbounded variable $t$ . We discuss the asymptotic behavior, as $t \to \infty$ , of solutions which are defined and bounded on $Ω \times (0, \infty)$ .

Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales

Tatiana Danielsson, Pernilla Johnsen (2021)

Mathematica Bohemica

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In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^{2} (0, T; H_{0}^{1} (Ω))$ , fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation $ε^{p} \partial_{t} u_{ε} (x, t) - \nabla \cdot (a (x ε^{- 1}, x ε^{- 2}, t ε^{- q}, t ε^{- r}) \nabla u_{ε} (x, t)) = f (x, t)$ , where $0 < p < q < r$ . The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by $p$ , compared to the standard matching that gives rise...

On the long-time behaviour of solutions of the p-Laplacian parabolic system

Paweł Goldstein (2008)

Colloquium Mathematicae

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Convergence of global solutions to stationary solutions for a class of degenerate parabolic systems related to the p-Laplacian operator is proved. A similar result is obtained for a variable exponent p. In the case of p constant, the convergence is proved to be $¹_{l o c}$ , and in the variable exponent case, L² and $W^{1, p (x)}$ -weak.

Asymptotically self-similar solutions for the parabolic system modelling chemotaxis

Yūki Naito (2006)

Banach Center Publications

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We consider a nonlinear parabolic system modelling chemotaxis $u_{t} = \nabla \cdot (\nabla u - u \nabla v)$ , $v_{t} = Δ v + u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

Existence results for some nonlinear parabolic equations with degenerate coercivity and singular lower-order terms

Rabah Mecheter, Fares Mokhtari (2023)

Mathematica Bohemica

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In this paper, we study the existence results for some parabolic equations with degenerate coercivity, singular lower order term depending on the gradient, and positive initial data in $L^{1}$ .

On admissibility for parabolic equations in ℝⁿ

Martino Prizzi (2003)

Fundamenta Mathematicae

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We consider the parabolic equation (P) $u_{t} - Δ u = F (x, u)$ , (t,x) ∈ ℝ₊ × ℝⁿ, and the corresponding semiflow π in the phase space H¹. We give conditions on the nonlinearity F(x,u), ensuring that all bounded sets of H¹ are π-admissible in the sense of Rybakowski. If F(x,u) is asymptotically linear, under appropriate non-resonance conditions, we use Conley’s index theory to prove the existence of nontrivial equilibria of (P) and of heteroclinic trajectories joining some of these equilibria. The results obtained...

The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi, Giuseppe Mingione (2014)

Journal of the European Mathematical Society

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The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of $p$ -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

Homogenization of monotone parabolic problems with an arbitrary number of spatial and temporal scales

Tatiana Danielsson, Liselott Flodén, Pernilla Johnsen, Marianne Olsson Lindberg (2024)

Applications of Mathematics

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We prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of the scale parameter $ε$ . The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evolution problems.

$L^{p}$ -decay of solutions to dissipative-dispersive perturbations of conservation laws

Grzegorz Karch (1997)

Annales Polonici Mathematici

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We study the decay in time of the spatial $L^{p}$ -norm (1 ≤ p ≤ ∞) of solutions to parabolic conservation laws with dispersive and dissipative terms added uₜ - uₓₓₜ - νuₓₓ + buₓ = f(u)ₓ or uₜ + uₓₓₓ - νuₓₓ + buₓ = f(u)ₓ, and we show that under general assumptions about the nonlinearity, solutions of the nonlinear equations have the same long time behavior as their linearizations at the zero solution.

Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on $ℝ^{N}$

Cung The Anh, Le Thi Thuy (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove the existence of global attractors for the following semilinear degenerate parabolic equation on $ℝ^{N}$ : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method. ...

Estimates of Green functions and their applications for parabolic operators with singular potentials

Lotfi Riahi (2003)

Colloquium Mathematicae

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We prove global pointwise estimates for the Green function of a parabolic operator with potential in the parabolic Kato class on a $C^{1, 1}$ cylindrical domain Ω. We apply these estimates to obtain a new and shorter proof of the Harnack inequality [16], and to study the boundary behavior of nonnegative solutions.

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Philippe Souplet, Slim Tayachi (2001)

Colloquium Mathematicae

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Consider the nonlinear heat equation (E): $u_{t} - Δ u = {| u |}^{p - 1} u + b {| \nabla u |}^{q}$ . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C ₁ {(T - t)}^{- 1 / (p - 1)} {\leq | | u (t) | |}_{\infty} \leq C ₂ {(T - t)}^{- 1 / (p - 1)}$ . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality $u_{t} - u_{x x} \geq u^{p}$ . More general inequalities of the form $u_{t} - u_{x x} \geq f (u)$ with, for instance, $f (u) = (1 + u) l o g^{p} (1 + u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...

On the Picard problem for hyperbolic differential equations in Banach spaces

Antoni Sadowski (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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B. Rzepecki in [5] examined the Darboux problem for the hyperbolic equation $z_{x y} = f (x, y, z, z_{x y})$ on the quarter-plane x ≥ 0, y ≥ 0 via a fixed point theorem of B.N. Sadovskii [6]. The aim of this paper is to study the Picard problem for the hyperbolic equation $z_{x y} = f (x, y, z, z_{x}, z_{x y})$ using a method developed by A. Ambrosetti [1], K. Goebel and W. Rzymowski [2] and B. Rzepecki [5].

Existence and stability results of nonlinear higher-order wave equation with a nonlinear source term and a delay term

Mama Abdelli, Abderrahmane Beniani, Nadia Mezouar, Ahmed Chahtou (2023)

Mathematica Bohemica

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We consider the initial-boundary value problem for a nonlinear higher-order nonlinear hyperbolic equation in a bounded domain. The existence of global weak solutions for this problem is established by using the potential well theory combined with Faedo-Galarkin method. We also established the asymptotic behavior of global solutions as $t \to \infty$ by applying the Lyapunov method.