Finite-dimensional pullback attractors for parabolic equations with Hardy type potentials

Cung The Anh; Ta Thi Hong Yen

Displaying similar documents to “Finite-dimensional pullback attractors for parabolic equations with Hardy type potentials”

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. ...

Porous medium equation and fast diffusion equation as gradient systems

Samuel Littig, Jürgen Voigt (2015)

Czechoslovak Mathematical Journal

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We show that the Porous Medium Equation and the Fast Diffusion Equation, $\dot{u} - Δ u^{m} = f$ , with $m \in (0, \infty)$ , can be modeled as a gradient system in the Hilbert space $H^{- 1} (Ω)$ , and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets $Ω \subseteq ℝ^{n}$ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.

$L^{p}$ -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.

Lyapunov functions and $L^{p}$ -estimates for a class of reaction-diffusion systems

Dirk Horstmann (2001)

Colloquium Mathematicae

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We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, $ε c ₜ = k_{c} Δ c - f (c) c + g (a, c)$ , x ∈ Ω, t > 0, for $Ω \subset ℝ^{N}$ , completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform $L^{p}$ -estimates.

On estimation of diffusion coefficient based on spatio-temporal FRAP images: An inverse ill-posed problem

Kaňa, Radek, Matonoha, Ctirad, Papáček, Štěpán, Soukup, Jindřich

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We present the method for determination of phycobilisomes diffusivity (diffusion coefficient $D$ ) on thylakoid membrane from fluorescence recovery after photobleaching (FRAP) experiments. This was usually done by analytical models consisting mainly of a simple curve fitting procedure. However, analytical models need some unrealistic conditions to be supposed. Our method, based on finite difference approximation of the process governed by the Fickian diffusion equation and on the minimization...

Existence and upper semicontinuity of uniform attractors in $H ¹ (ℝ^{N})$ for nonautonomous nonclassical diffusion equations

Cung The Anh, Nguyen Duong Toan (2014)

Annales Polonici Mathematici

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We prove the existence of uniform attractors $_{ε}$ in the space $H ¹ (ℝ^{N})$ for the nonautonomous nonclassical diffusion equation $u_{t} - ε Δ u_{t} - Δ u + f (x, u) + λ u = g (x, t)$ , ε ∈ [0,1]. The upper semicontinuity of the uniform attractors ${_{ε}}_{ε \in [0, 1]}$ at ε = 0 is also studied.

Attractors for stochastic reaction-diffusion equation with additive homogeneous noise

Jakub Slavík (2021)

Czechoslovak Mathematical Journal

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We study the asymptotic behaviour of solutions of a reaction-diffusion equation in the whole space $ℝ^{d}$ driven by a spatially homogeneous Wiener process with finite spectral measure. The existence of a random attractor is established for initial data in suitable weighted $L^{2}$ -space in any dimension, which complements the result from P. W. Bates, K. Lu, and B. Wang (2013). Asymptotic compactness is obtained using elements of the method of short trajectories.

Global existence and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable exponents

Aya Khaldi, Amar Ouaoua, Messaoud Maouni (2022)

Mathematica Bohemica

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We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms $u_{t} - M (\int_{Ω} {| \nabla u |}^{2} d x) Δ u + {| u |}^{m (x) - 2} u_{t} = {| u |}^{r (x) - 2} u .$ We prove with suitable assumptions on the variable exponents $r (\cdot),$ $m (\cdot)$ the global existence of the solution and a stability result using potential and Nihari’s functionals with small positive initial energy, the stability being based on Komornik’s inequality.

On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains

M. Prizzi, K. P. Rybakowski (2003)

Studia Mathematica

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We study a family of semilinear reaction-diffusion equations on spatial domains $Ω_{ε}$ , ε > 0, in $ℝ^{l}$ lying close to a k-dimensional submanifold ℳ of $ℝ^{l}$ . As ε → 0⁺, the domains collapse onto (a subset of) ℳ. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by $H ¹_{s} (Ω)$ . The definition of $H ¹_{s} (Ω)$ , given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable...

Existence and asymptotic behaviour of some time-inhomogeneous diffusions

Mihai Gradinaru, Yoann Offret (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient ${ρ sgn (x) | x |}^{α} / t^{β}$ . This process can be viewed as a Brownian motion evolving in a potential, possibly singular, depending on time. We prove results on the existence and uniqueness of solution, study its asymptotic behaviour and made a precise description, in terms of parameters $ρ$ , $α$ and $β$ , of the recurrence, transience and convergence. More precisely,...

Global existence and $L_{p}$ decay estimate of solution for Cahn-Hilliard equation with inertial term

Hongmei Xu, Qi Li (2021)

Applications of Mathematics

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The Cauchy problem of the Cahn-Hilliard equation with inertial term in multi space dimension is considered. Based on detailed analysis of Green’s function, using fixed-point theorem, we get the global existence in time of classical solution with large initial data. Furthermore, we get $L_{p}$ decay rate of the solution.

Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda (2002)

Colloquium Mathematicae

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We study the absence of nonnegative global solutions to parabolic inequalities of the type $u_{t} \geq - {(- Δ)}^{β / 2} u - V (x) u + h (x, t) u^{p}$ , where ${(- Δ)}^{β / 2}$ , 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying ${V ₊ (x) \sim a | x |}^{- b}$ , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....

Bi-spaces global attractors in abstract parabolic equations

J. W. Cholewa, T. Dłotko (2003)

Banach Center Publications

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An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on $X^{α}$ . This semigroup possesses an $(X^{α} - Z)$ -global attractor that is closed, bounded, invariant in $X^{α}$ , and attracts bounded subsets of $X^{α}$ in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system. ...

Asymptotic behavior of a sequence defined by iteration with applications

Stevo Stević (2002)

Colloquium Mathematicae

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We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f (x, y) = p x + (1 - p) y - \sum_{s = m}^{\infty} s (x, y)$ uniformly in a neighborhood of the origin, where m > 1, $_{s} (x, y) = \sum_{i = 0}^{s} a_{i, s} x^{s - i} y^{i}$ ; (c) $ₘ (1, 1) = \sum_{i = 0}^{m} a_{i, m} > 0$ . Let x₀,x₁ ∈ (0,α) and $x_{n + 1} = f (x ₙ, x_{n - 1})$ , n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: $x ₙ \sim {((2 - p) / ((m - 1) \sum_{i = 0}^{m} a_{i, m}))}^{1 / (m - 1)} 1 / \sqrt[m - 1]{n}$ .

Infinite Iterated Function Systems Depending on a Parameter

Ludwik Jaksztas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets $J_{0, σ}$ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of $J_{0, σ}$ , given by Urbański and Zinsmeister. The closure of the limit set of our IFS $ϕ_{σ, α}^{n, k}$ is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...

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.

Weak- $L^{p}$ solutions for a model of self-gravitating particles with an external potential

Andrzej Raczyński (2007)

Studia Mathematica

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The existence of solutions to a nonlinear parabolic equation describing the temporal evolution of a cloud of self-gravitating particles with a given external potential is studied in weak- $L^{p}$ spaces (i.e. Markiewicz spaces). The main goal is to prove the existence of global solutions and to study their large time behaviour.

Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times

Anton Bovier, Michael Eckhoff, Véronique Gayrard, Markus Klein (2004)

Journal of the European Mathematical Society

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We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form $- ϵ Δ + \nabla F (\cdot) \nabla$ on $ℝ^{d}$ or subsets of $ℝ^{d}$ , where $F$ is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of $F$ can be related, up to multiplicative errors that tend to one as $ϵ ↓ 0$ , to the capacities of suitably constructed sets. We show that...

Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues

Anton Bovier, Véronique Gayrard, Markus Klein (2005)

Journal of the European Mathematical Society

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We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form $- ϵ Δ + \nabla F (\cdot) \nabla$ on $ℝ^{d}$ or subsets of $ℝ^{d}$ , where $F$ is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of...

Attractor of a semi-discrete Benjamin-Bona-Mahony equation on ℝ¹

Chaosheng Zhu (2015)

Annales Polonici Mathematici

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This paper is concerned with the study of the large time behavior and especially the regularity of the global attractor for the semi-discrete in time Crank-Nicolson scheme to discretize the Benjamin-Bona-Mahony equation on ℝ¹. Firstly, we prove that this semi-discrete equation provides a discrete infinite-dimensional dynamical system in H¹(ℝ¹). Then we prove that this system possesses a global attractor $_{τ}$ in H¹(ℝ¹). In addition, we show that the global attractor $_{τ}$ is regular, i.e., $_{τ}$ ...

Single-point blow-up for a semilinear parabolic system

Ph. Souplet (2009)

Journal of the European Mathematical Society

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We consider positive solutions of the system $u_{t} - Δ u = v^{p}$ ; $v_{t} - Δ v = u^{q}$ in a ball or in the whole space, with $p, q > 1$ . Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case $p = q$ . Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for...

Blow up for a completely coupled Fujita type reaction-diffusion system

Noureddine Igbida, Mokhtar Kirane (2002)

Colloquium Mathematicae

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This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $u ₜ - Δ (a_{11} u) = h (t, x) {| v |}^{p}$ , $v ₜ - Δ (a_{21} u) - Δ (a_{22} v) = k (t, x) {| w |}^{q}$ , $w ₜ - Δ (a_{31} u) - Δ (a_{32} v) - Δ (a_{33} w) = l (t, x) {| u |}^{r}$ , for $x \in ℝ^{N}$ , t > 0, p > 0, q > 0, r > 0, $a_{i j} = a_{i j} (t, x, u, v)$ , under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for $x \in ℝ^{N}$ , where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the...

Envelope functions and asymptotic structures in Banach spaces

Bünyamin Sarı (2004)

Studia Mathematica

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We introduce a notion of disjoint envelope functions to study asymptotic structures of Banach spaces. The main result gives a new characterization of asymptotic- $ℓ_{p}$ spaces in terms of the $ℓ_{p}$ -behavior of “disjoint-permissible” vectors of constant coefficients. Applying this result to Tirilman spaces we obtain a negative solution to a conjecture of Casazza and Shura. Further investigation of the disjoint envelopes leads to a finite-representability result in the spirit of the Maurey-Pisier...