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

M. Prizzi; K. P. Rybakowski

Displaying similar documents to “On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains”

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...

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.

Curved thin domains and parabolic equations

M. Prizzi, M. Rinaldi, K. P. Rybakowski (2002)

Studia Mathematica

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Consider the family uₜ = Δu + G(u), t > 0, $x \in Ω_{ε}$ , $\partial_{ν_{ε}} u = 0$ , t > 0, $x \in \partial Ω_{ε}$ , $(E_{ε})$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set $Ω_{ε}$ is a thin domain in $ℝ^{l}$ , possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of $ℝ^{l}$ . If G is dissipative, then equation $(E_{ε})$ has a global attractor $_{ε}$ . We identify a “limit” equation for the family $(E_{ε})$ , prove convergence of trajectories and establish an upper semicontinuity result for the family $_{ε}$ as ε → 0⁺. ...

On n-circled $ℋ^{\infty}$ -domains of holomorphy

Marek Jarnicki, Peter Pflug (1997)

Annales Polonici Mathematici

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We present various characterizations of n-circled domains of holomorphy $G \subset ℂ^{n}$ with respect to some subspaces of $ℋ^{\infty} (G)$ .

The $L^{2}$ $\bar{\partial}$ -Cauchy problem on weakly $q$ -pseudoconvex domains in Stein manifolds

Sayed Saber (2015)

Czechoslovak Mathematical Journal

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Let $X$ be a Stein manifold of complex dimension $n \geq 2$ and $Ω ⋐ X$ be a relatively compact domain with $C^{2}$ smooth boundary in $X$ . Assume that $Ω$ is a weakly $q$ -pseudoconvex domain in $X$ . The purpose of this paper is to establish sufficient conditions for the closed range of $\bar{\partial}$ on $Ω$ . Moreover, we study the $\bar{\partial}$ -problem on $Ω$ . Specifically, we use the modified weight function method to study the weighted $\bar{\partial}$ -problem with exact support in $Ω$ . Our method relies on the $L^{2}$ -estimates by Hörmander (1965) and by Kohn (1973). ...

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.

Pluriharmonic extension in proper image domains

Rafał Czyż (2009)

Annales Polonici Mathematici

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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D ₁ \times \dots \times D_{s}$ , j=1,...,s, s ≥ 3. Also let $Ω_{π}$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f : \partial Ω_{π} \to ℝ$ that can be extended to a real-valued pluriharmonic function in $Ω_{π}$ .

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...

A domain whose envelope of holomorphy is not a domain

Edgar Lee Stout (2006)

Annales Polonici Mathematici

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We construct a domain of holomorphy in $ℂ^{N}$ , N≥ 2, whose envelope of holomorphy is not diffeomorphic to a domain in $ℂ^{N}$ .

On the paradox of two $R_{1}, R_{2}$ domains in Schwarzschild exterior solution

A. Zięba (1972)

Colloquium Mathematicae

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Characterization of cycle domains via Kobayashi hyperbolicity

Gregor Fels, Alan Huckleberry (2005)

Bulletin de la Société Mathématique de France

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A real form $G$ of a complex semi-simple Lie group $G^{ℂ}$ has only finitely many orbits in any given $G^{ℂ}$ -flag manifold $Z = G^{ℂ} / Q$ . The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits $D$ generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of $D$ which, with very few well-understood exceptions, are parameterized by the Wolf cycle...

A remark on separate holomorphy

Marek Jarnicki, Peter Pflug (2006)

Studia Mathematica

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Let X be a Riemann domain over $ℂ^{k} \times ℂ^{ℓ}$ . If X is a domain of holomorphy with respect to a family ℱ ⊂(X), then there exists a pluripolar set $P \subset ℂ^{k}$ such that every slice $X_{a}$ of X with a∉ P is a region of holomorphy with respect to the family ${f |}_{X_{a}} : f \in ℱ$ .

Rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds

Stephen S.-T. Yau (2011)

Journal of the European Mathematical Society

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Let $X_{1}$ and $X_{2}$ be two compact strongly pseudoconvex CR manifolds of dimension $2 n - 1 \geq 5$ which bound complex varieties $V_{1}$ and $V_{2}$ with only isolated normal singularities in $ℂ^{N 1}$ and $ℂ^{N 2}$ respectively. Let $S_{1}$ and $S_{2}$ be the singular sets of $V_{1}$ and $V_{2}$ respectively and $S_{2}$ is nonempty. If $2 n - N_{2} - 1 \geq 1$ and the cardinality of $S_{1}$ is less than 2 times the cardinality of $S_{2}$ , then we prove that any non-constant CR morphism from $X_{1}$ to $X_{2}$ is necessarily a CR biholomorphism. On the other hand, let $X$ be a compact strongly pseudoconvex CR...

Steinness of bundles with fiber a Reinhardt bounded domain

Karl Oeljeklaus, Dan Zaffran (2006)

Bulletin de la Société Mathématique de France

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Let $E$ denote a holomorphic bundle with fiber $D$ and with basis $B$ . Both $D$ and $B$ are assumed to be Stein. For $D$ a Reinhardt bounded domain of dimension $d = 2$ or $3$ , we give a necessary and sufficient condition on $D$ for the existence of a non-Stein such $E$ (Theorem $1$ ); for $d = 2$ , we give necessary and sufficient criteria for $E$ to be Stein (Theorem $2$ ). For $D$ a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for $E$ to be Stein (Theorem...

From a kinetic equation to a diffusion under an anomalous scaling

Giada Basile (2014)

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

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A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $(K (t), i (t), Y (t))$ on $(𝕋^{2} \times {1, 2} \times ℝ^{2})$ , where $𝕋^{2}$ is the two-dimensional torus. Here $(K (t), i (t))$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y (t)$ is an additive functional of $K$ , defined as $\int_{0}^{t} v (K (s)) d s$ , where $| v | \sim 1$ for small $k$ . We prove that the rescaled process ${(N ln N)}^{- 1 / 2} Y (N t)$ converges in distribution to a two-dimensional Brownian motion. As a consequence,...

When $C_{p} (X)$ is domain representable

William Fleissner, Lynne Yengulalp (2013)

Fundamenta Mathematicae

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Let M be a metrizable group. Let G be a dense subgroup of $M^{X}$ . We prove that if G is domain representable, then $G = M^{X}$ . The following corollaries answer open questions. If X is completely regular and $C_{p} (X)$ is domain representable, then X is discrete. If X is zero-dimensional, T₂, and $C_{p} (X,)$ is subcompact, then X is discrete.

The method of rotation and Marcinkiewicz integrals on product domains

Jiecheng Chen, Dashan Fan, Yiming Ying (2002)

Studia Mathematica

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We give some rather weak sufficient condition for $L^{p}$ boundedness of the Marcinkiewicz integral operator $μ_{Ω}$ on the product spaces $ℝ ⁿ \times ℝ^{m}$ (1 < p < ∞), which improves and extends some known results.

Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities

Boumediene Abdellaoui, Eduardo Colorado, Ireneo Peral (2004)

Journal of the European Mathematical Society

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In this work we study the problem $u_{t} {- div (| x |}^{- 2 γ} \nabla u) = λ \frac{u^{α}}{{| x |}^{2 (γ + 1)}} + f$ in $Ω \times (0, T)$ , $u \geq 0$ in $Ω \times (0, T)$ , $u = 0$ on $\partial Ω \times (0, T)$ , $u (x, 0) = u_{0} (x)$ in $Ω$ , $Ω \subset ℝ^{N}$ $(N \geq 2)$ is a bounded regular domain such that $0 \in Ω$ , $λ > 0$ , $α > 0$ , $- \infty < γ < (N - 2) / 2$ , $f$ and $u_{0}$ are positive functions such that $f \in L^{1} (Ω \times (0, T))$ and $u_{0} \in L^{1} (Ω)$ . The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $α = 1$ , $1 + γ > 0$ ; (ii) the nonexistence of solutions if $α > 1$ , $1 + γ > 0$ ; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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We study the chemotaxis system with singular sensitivity and logistic-type source: $u_{t} = Δ u - χ \nabla \cdot (u \nabla v / v) + r u - μ u^{k}$ , $0 = Δ v - v + u$ under the non-flux boundary conditions in a smooth bounded domain $Ω \subset ℝ^{n}$ , $χ, r, μ > 0$ , $k > 1$ and $n \geq 1$ . It is shown with $k \in (1, 2)$ that the system possesses a global generalized solution for $n \geq 2$ which is bounded when $χ > 0$ is suitably small related to $r > 0$ and the initial datum is properly small, and a global bounded classical solution for $n = 1$ .

On two-to-one continuous functions

J. Mioduszewski

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CONTENTSIntroduction................................................................................................................................................................................3I. General properties of k-to-one functions on locally compact spaces1. Multi-valued functions Ф and ψ......................................................................................................................................... 62. The proof of (I.11)..................................................................................................................................................................

On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let $ℱ_{h}^{i} (k, n)$ be the $i$ -th ordered configuration space of all distinct points $H_{1}, ..., H_{h}$ in the Grassmannian $G r (k, n)$ of $k$ -dimensional subspaces of $ℂ^{n}$ , whose sum is a subspace of dimension $i$ . We prove that $ℱ_{h}^{i} (k, n)$ is (when non empty) a complex submanifold of $G r {(k, n)}^{h}$ of dimension $i (n - i) + h k (i - k)$ and its fundamental group is trivial if $i = m i n (n, h k)$ , $h k \neq n$ and $n > 2$ and equal to the braid group of the sphere $ℂ$ $P^{1}$ if $n = 2$ . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k = n - 1$ .

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let $Ω$ be a bounded domain of class $C^{2}$ in $ℝ$ N and let $K$ be a compact subset of $\partial Ω$ . Assume that $q \geq (N + 1) / (N - 1)$ and denote by $U_{K}$ the maximal solution of $- Δ u + u^{q} = 0$ in $Ω$ which vanishes on $\partial Ω ∖ K$ . We obtain sharp upper and lower estimates for $U_{K}$ in terms of the Bessel capacity $C_{2 / q, q^{'}}$ and prove that $U_{K}$ is $σ$ -moderate. In addition we describe the precise asymptotic behavior of $U_{K}$ at points $σ \in K$ , which depends on the “density” of $K$ at $σ$ , measured in terms of the capacity $C_{2 / q, q^{'}}$ .

Local density of diffeomorphisms with large centralizers

Christian Bonatti, Sylvain Crovisier, Gioia M. Vago, Amie Wilkinson (2008)

Annales scientifiques de l'École Normale Supérieure

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Given any compact manifold $M$ , we construct a non-empty open subset $𝒪$ of the space ${Diff}^{1} (M)$ of $C^{1}$ -diffeomorphisms and a dense subset $𝒟 \subset 𝒪$ such that the centralizer of every diffeomorphism in $𝒟$ is uncountable, hence non-trivial.

On the Picard number of divisors in Fano manifolds

Cinzia Casagrande (2012)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$ . We consider the image $𝒩_{1} (D, X)$ of $𝒩_{1} (D)$ in $𝒩_{1} (X)$ under the natural push-forward of $1$ -cycles. We show that $ρ_{X} - ρ_{D} \leq codim 𝒩_{1} (D, X) \leq 8$ . Moreover if $codim 𝒩_{1} (D, X) \geq 3$ , then either $X ≅ S \times T$ where $S$ is a Del Pezzo surface, or $codim 𝒩_{1} (D, X) = 3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that $ρ_{X} - ρ_{T} = 4$ .

Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

Catherine Bandle, Joachim von Below, Wolfgang Reichel (2008)

Journal of the European Mathematical Society

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We consider linear elliptic equations $- Δ u + q (x) u = λ u + f$ in bounded Lipschitz domains $D \subset ℝ^{N}$ with mixed boundary conditions $\partial u / \partial n = σ (x) λ u + g$ on $\partial D$ . The main feature of this boundary value problem is the appearance of $λ$ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient $σ (x)$ . We study positivity principles and anti-maximum principles. One of our main results states that if $σ$ is somewhere negative, $q \geq 0$ and $\int_{D} q (x) d x > 0$ then there exist two eigenvalues $λ_{- 1}$ , $λ_{1}$ such the positivity...