Displaying similar documents to “Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux”

On the uniqueness of periodic decomposition

Viktor Harangi (2011)

Fundamenta Mathematicae

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Let a , . . . , a k be arbitrary nonzero real numbers. An ( a , . . . , a k ) -decomposition of a function f:ℝ → ℝ is a sum f + + f k = f where f i : is an a i -periodic function. Such a decomposition is not unique because there are several solutions of the equation h + + h k = 0 with h i : a i -periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the ( a , . . . , a k ) -decomposition is essentially unique. We characterize those periods for which essential...

The regularity of the positive part of functions in L 2 ( I ; H 1 ( Ω ) ) H 1 ( I ; H 1 ( Ω ) * ) with applications to parabolic equations

Daniel Wachsmuth (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let u L 2 ( I ; H 1 ( Ω ) ) with t u L 2 ( I ; H 1 ( Ω ) * ) be given. Then we show by means of a counter-example that the positive part u + of u has less regularity, in particular it holds t u + L 1 ( I ; H 1 ( Ω ) * ) in general. Nevertheless, u + satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

Cobham's theorem for substitutions

Fabien Durand (2011)

Journal of the European Mathematical Society

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The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let α and β be two multiplicatively independent Perron numbers. Then a sequence x A , where A is a finite alphabet, is both α -substitutive and β -substitutive if and only if x is ultimately...

On higher-order semilinear parabolic equations with measures as initial data

Victor Galaktionov (2004)

Journal of the European Mathematical Society

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We consider 2 m th-order ( m 2 ) semilinear parabolic equations u t = ( Δ ) m u ± | u | p 1 u in N × + ( p > 1 ) , with Dirac’s mass δ ( x ) as the initial function. We show that for p < p 0 = 1 + 2 m / N , the Cauchy problem admits a solution u ( x , t ) which is bounded and smooth for small t > 0 , while for p p 0 such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.

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 u ) = λ u α | x | 2 ( γ + 1 ) + f in Ω × ( 0 , T ) , u 0 in Ω × ( 0 , T ) , u = 0 on Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , Ω N ( N 2 ) is a bounded regular domain such that 0 Ω , λ > 0 , α > 0 , - < γ < ( N 2 ) / 2 , f and u 0 are positive functions such that f L 1 ( Ω × ( 0 , T ) ) and u 0 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 behavior of the difference equation x n + 1 = a x n - 3 b + c x n - 1 x n - 3

Raafat Abo-Zeid (2015)

Archivum Mathematicum

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In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation x n + 1 = a x n - 3 b + c x n - 1 x n - 3 , n = 0 , 1 , where a , b , c are positive real numbers and the initial conditions x - 3 , x - 2 , x - 1 , x 0 are real numbers.

Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco, Patrícia Gonçalves, Adriana Neumann (2013)

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

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We consider the exclusion process in the one-dimensional discrete torus with N points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance N - β , with β [ 0 , ) . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter β . If β [ 0 , 1 ) , the hydrodynamic limit is given by the usual heat equation. If β = 1 , it is given by a parabolic equation involving an operator...

The Lebesgue constant for the periodic Franklin system

Markus Passenbrunner (2011)

Studia Mathematica

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We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots t j = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let V n , ν be the space of piecewise linear continuous functions on the torus with knots t j : 0 j N - 1 . Finally, let P n , ν be the orthogonal projection operator from L²([0,1)) onto V n , ν . The main result is l i m n , ν = 1 | | P n , ν : L L | | = s u p n , 0 ν n | | P n , ν : L L | | = 2 + ( 33 - 18 3 ) / 13 . This shows in particular that the Lebesgue constant of the classical...

On a new normalization for tractor covariant derivatives

Matthias Hammerl, Petr Somberg, Vladimír Souček, Josef Šilhan (2012)

Journal of the European Mathematical Society

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A regular normal parabolic geometry of type G / P on a manifold M gives rise to sequences D i of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ω on the corresponding tractor bundle V , where ω is the normal Cartan connection. The first operator D 0 in the sequence is overdetermined and it is well known that ω yields the prolongation of this operator in the homogeneous case M = G / P . Our first...

C 1 self-maps on closed manifolds with finitely many periodic points all of them hyperbolic

Jaume Llibre, Víctor F. Sirvent (2016)

Mathematica Bohemica

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Let X be a connected closed manifold and f a self-map on X . We say that f is almost quasi-unipotent if every eigenvalue λ of the map f * k (the induced map on the k -th homology group of X ) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of λ as eigenvalue of all the maps f * k with k odd is equal to the sum of the multiplicities of λ as eigenvalue of all the maps f * k with k even. We prove that if f is C 1 having finitely many periodic points all of them...

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 · ( u v / v ) + r u - μ u k , 0 = Δ v - v + u under the non-flux boundary conditions in a smooth bounded domain Ω n , χ , r , μ > 0 , k > 1 and n 1 . It is shown with k ( 1 , 2 ) that the system possesses a global generalized solution for n 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 .

Partial Hölder continuity for quasilinear parabolic systems of higher order with strictly controlled growth

Mario Marino, Antonino Maugeri (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Sfruttando i risultati di [1], si prova che le derivate spaziali D α u di ordine | α | con | α | < m - 1 delle soluzioni in Q di un sistema parabolico quasilineare di ordine 2 m con andamenti strettamente controllati, sono parzialmente hölderiane in Q con esponente di hölderianità decrescente al crescere di | α | .

Covariance structure of wide-sense Markov processes of order k ≥ 1

Arkadiusz Kasprzyk, Władysław Szczotka (2006)

Applicationes Mathematicae

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A notion of a wide-sense Markov process X t of order k ≥ 1, X t W M ( k ) , is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of X t is the k-dimensional process x t = ( X t - k + 1 , . . . , X t ) . The covariance structure of X t W M ( k ) is considered in the general case and in the periodic case. In the general case it is shown that X t W M ( k ) iff x t is a k-dimensional WM(1) process and iff the covariance function of x t has the triangular...

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 Ω ε , ν ε u = 0 , t > 0, x Ω ε , ( E ε ) of semilinear Neumann boundary value problems, where, for ε > 0 small, the set Ω ε is a thin domain in l , possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of 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⁺. ...

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 N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on 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 K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials P c ( z ) = z d + c for c ∈ ℂ. Denote by J c the Julia set of P c and let d = c | J c i s c o n n e c t e d be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c d : those for which the critical point 0 is not recurrent by P c and without parabolic cycles. The Hausdorff dimension of J c , denoted by H D ( J c ) , does not depend continuously on c at such c d ; on the other hand the function c H D ( J c ) is analytic in - d . Our first result asserts that there is still some...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Weak convergence of mutually independent X B and X A under weak convergence of X X B - X A

W. Szczotka (2006)

Applicationes Mathematicae

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For each n ≥ 1, let v n , k , k 1 and u n , k , k 1 be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let X B ( t ) = ( 1 / c ) j = 1 [ n t ] ( v n , j - v ̅ ) , X A ( t ) = ( 1 / c ) j = 1 [ n t ] ( u n , j - u ̅ ̅ ) , t ≥ 0, and X = X B - X A . The main result gives conditions under which the weak convergence X X , where X is a Lévy process, implies X B X B and X A X A , where X B and X A are mutually independent Lévy processes and X = X B - X A .

Nonexistence results for the Cauchy problem of some systems of hyperbolic equations

Mokhtar Kirane, Salim Messaoudi (2002)

Annales Polonici Mathematici

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We consider the systems of hyperbolic equations ⎧ u = Δ ( a ( t , x ) u ) + Δ ( b ( t , x ) v ) + h ( t , x ) | v | p , t > 0, x N , (S1) ⎨ ⎩ v = Δ ( c ( t , x ) v ) + k ( t , x ) | u | q , t > 0, x N u = Δ ( a ( t , x ) u ) + h ( t , x ) | v | p , t > 0, x N , (S2) ⎨ ⎩ v = Δ ( c ( t , x ) v ) + l ( t , x ) | v | m + k ( t , x ) | u | q , t > 0, x N , (S3) ⎧ u = Δ ( a ( t , x ) u ) + Δ ( b ( t , x ) v ) + h ( t , x ) | u | p , t > 0, x N , ⎨ ⎩ v = Δ ( c ( t , x ) v ) + k ( t , x ) | v | q , t > 0, x N , in ( 0 , ) × N with u(0,x) = u₀(x), v(0,x) = v₀(x), uₜ(0,x) = u₁(x), vₜ(0,x) = v₁(x). We show that, in each case, there exists a bound B on N such that for 1 ≤ N ≤ B solutions to the systems blow up in finite time.