Attractors for stochastic reaction-diffusion equation with additive homogeneous noise

Jakub Slavík

Displaying similar documents to “Attractors for stochastic reaction-diffusion equation with additive homogeneous noise”

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.

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...

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.

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...

Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed

Marek Fila, Michael Winkler (2008)

Journal of the European Mathematical Society

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We consider a reaction-diffusion-convection equation on the halfline (0,1) with the zero Dirichlet boundary condition at $x = 0$ . We find a positive selfsimilar solution $u$ which blows up in a finite time $T$ at $x = 0$ while $u (x, T)$ remains bounded for $x > 0$ .

Regularity of the effective diffusivity of random diffusion with respect to anisotropy coefficient

M. Cudna, T. Komorowski (2008)

Studia Mathematica

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We show that the effective diffusivity of a random diffusion with a drift is a continuous function of the drift coefficient. In fact, in the case of a homogeneous and isotropic random environment the function is $C^{\infty}$ smooth outside the origin. We provide a one-dimensional example which shows that the diffusivity coefficient need not be differentiable at 0.

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...

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...

Quasi-diffusion solution of a stochastic differential equation

Agnieszka Plucińska, Wojciech Szymański (2007)

Applicationes Mathematicae

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We consider the stochastic differential equation $X_{t} = X ₀ + \int_{0}^{t} (A_{s} + B_{s} X_{s}) d s + \int_{0}^{t} C_{s} d Y_{s}$ , where $A_{t}$ , $B_{t}$ , $C_{t}$ are nonrandom continuous functions of t, X₀ is an initial random variable, $Y = (Y_{t}, t \geq 0)$ is a Gaussian process and X₀, Y are independent. We give the form of the solution ( $X_{t}$ ) to (0.1) and then basing on the results of Plucińska [Teor. Veroyatnost. i Primenen. 25 (1980)] we prove that ( $X_{t}$ ) is a quasi-diffusion proces.

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...

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.

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.

Random walks in ${(ℤ_{+})}^{2}$ with non-zero drift absorbed at the axes

Irina Kurkova, Kilian Raschel (2011)

Bulletin de la Société Mathématique de France

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Spatially homogeneous random walks in ${(ℤ_{+})}^{2}$ with non-zero jump probabilities at distance at most $1$ , with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes. ...

Spreading and vanishing in nonlinear diffusion problems with free boundaries

Yihong Du, Bendong Lou (2015)

Journal of the European Mathematical Society

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We study nonlinear diffusion problems of the form $u_{t} = u_{x x} + f (u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f (u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f (u)$ which is $C^{1}$ and satisfies $f (0) = 0$ , we show that the omega limit set $ω (u)$ of every bounded positive solution is determined by a stationary...

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,...

Self-similar solutions in reaction-diffusion systems

Joanna Rencławowicz (2003)

Banach Center Publications

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In this paper we examine self-similar solutions to the system $u_{i t} - d_{i} Δ u_{i} = \prod_{k = 1}^{m} u_{k}^{p_{k}^{i}}$ , i = 1,…,m, $x \in ℝ^{N}$ , t > 0, $u_{i} (0, x) = u_{0 i} (x)$ , i = 1,…,m, $x \in ℝ^{N}$ , where m > 1 and $p_{k}^{i} > 0$ , to describe asymptotics near the blow up point.

Regularity of stopping times of diffusion processes in Besov spaces

Xicheng Zhang (2002)

Studia Mathematica

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We prove that the exit times of diffusion processes from a bounded open set Ω almost surely belong to the Besov space $B_{p, q}^{α} (Ω)$ provided that pα < 1 and 1 ≤ q < ∞.

Nonlinear diffusion equations with perturbation terms on unbounded domains

Kurima, Shunsuke

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This paper considers the initial-boundary value problem for the nonlinear diffusion equation with the perturbation term $u_{t} + (- Δ + 1) β (u) + G (u) = g in Ω \times (0, T)$ in an unbounded domain $Ω \subset ℝ^{N}$ with smooth bounded boundary, where $N \in ℕ$ , $T > 0$ , $β$ , is a single-valued maximal monotone function on $ℝ$ , e.g., $β (r) = {| r |}^{q - 1} r (q > 0, q \neq 1)$ and $G$ is a function on $ℝ$ which can be regarded as a Lipschitz continuous operator from ${(H^{1} (Ω))}^{*}$ to ${(H^{1} (Ω))}^{*}$ . The present work establishes existence and estimates for the above problem.

Inverse source problem in a space fractional diffusion equation from the final overdetermination

Amir Hossein Salehi Shayegan, Reza Bayat Tajvar, Alireza Ghanbari, Ali Safaie (2019)

Applications of Mathematics

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We consider the problem of determining the unknown source term $f = f (x, t)$ in a space fractional diffusion equation from the measured data at the final time $u (x, T) = ψ (x)$ . In this way, a methodology involving minimization of the cost functional $J (f) = \int_{0}^{l} {(u (x, t; f) |_{t = T} - ψ (x))}^{2} d x$ is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and...