Displaying similar documents to “Dynamics of a modified Davey-Stewartson system in ℝ³”

The n -centre problem of celestial mechanics for large energies

Andreas Knauf (2002)

Journal of the European Mathematical Society

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We consider the classical three-dimensional motion in a potential which is the sum of n attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the n centres, we find a universal behaviour for all energies E above a positive threshold. Whereas for n = 1 there are no bounded orbits, and for n = 2 there is just one closed orbit, for n 3 the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of...

Local energy decay for several evolution equations on asymptotically euclidean manifolds

Jean-François Bony, Dietrich Häfner (2012)

Annales scientifiques de l'École Normale Supérieure

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Let  P be a long range metric perturbation of the Euclidean Laplacian on  d , d 2 . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to  P . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group e i t f ( P ) where f has a suitable development at zero (resp. infinity). ...

Divergent solutions to the 5D Hartree equations

Daomin Cao, Qing Guo (2011)

Colloquium Mathematicae

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We consider the Cauchy problem for the focusing Hartree equation i u t + Δ u + ( | · | - 3 | u | ² ) u = 0 in ℝ⁵ with initial data in H¹, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of - Q + Δ Q + ( | · | - 3 | Q | ² ) Q = 0 in ℝ⁵, we prove that if u₀ ∈ H¹ satisfies M(u₀)E(u₀) < M(Q)E(Q) and ||∇u₀||₂||u₀||₂ > ||∇Q||₂||Q||₂, then the corresponding solution u(t) either blows up in finite forward time, or exists globally for positive time and there exists a time sequence tₙ → ∞ such that ||∇u(tₙ)||₂...

Scattering for 1D cubic NLS and singular vortex dynamics

Valeria Banica, Luis Vega (2012)

Journal of the European Mathematical Society

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We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions χ a ( t , x ) form a family of evolving regular curves in 3 that develop a singularity in finite time, indexed by a parameter a > 0 . We consider curves that are small regular perturbations of χ a ( t 0 , x ) for a fixed time t 0 . In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of...

A new look at an old comparison theorem

Jaroslav Jaroš (2021)

Archivum Mathematicum

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We present an integral comparison theorem which guarantees the global existence of a solution of the generalized Riccati equation on the given interval [ a , b ) when it is known that certain majorant Riccati equation has a global solution on [ a , b ) .

Global well-posedness for the Klein-Gordon-Schrödinger system with higher order coupling

Agus Leonardi Soenjaya (2022)

Mathematica Bohemica

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Global well-posedness for the Klein-Gordon-Schrödinger system with generalized higher order coupling, which is a system of PDEs in two variables arising from quantum physics, is proven. It is shown that the system is globally well-posed in ( u , n ) L 2 × L 2 under some conditions on the nonlinearity (the coupling term), by using the L 2 conservation law for u and controlling the growth of n via the estimates in the local theory. In particular, this extends the well-posedness results for such a system in...

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

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We study the Cauchy problem for the 3D MHD system with damping terms ε | u | α - 1 u and δ | b | β - 1 b (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

Grzegorz Karch (2000)

Studia Mathematica

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Large time behavior of solutions to the generalized damped wave equation u t t + A u t + ν B u + F ( x , t , u , u t , u ) = 0 for ( x , t ) n × [ 0 , ) is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where A u t = u t , B u = - Δ u , and the nonlinear term is either | u t | q - 1 u t or | u | α - 1 u . In this case, the asymptotic profile of solutions...

Total blow-up of a quasilinear heat equation with slow-diffusion for non-decaying initial data

Amy Poh Ai Ling, Masahiko Shimojō (2019)

Mathematica Bohemica

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We consider solutions of quasilinear equations u t = Δ u m + u p in N with the initial data u 0 satisfying 0 < u 0 < M and lim | x | u 0 ( x ) = M for some constant M > 0 . It is known that if 0 < m < p with p > 1 , the blow-up set is empty. We find solutions u that blow up throughout N when m > p > 1 .

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 = k = 1 m u k p k i , i = 1,…,m, x N , t > 0, u i ( 0 , x ) = u 0 i ( x ) , i = 1,…,m, x N , where m > 1 and p k i > 0 , to describe asymptotics near the blow up point.

Large data local solutions for the derivative NLS equation

Ioan Bejenaru, Daniel Tataru (2008)

Journal of the European Mathematical Society

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We consider the derivative NLS equation with general quadratic nonlinearities. In [2] the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension n = 2 . Here we prove a similar result for large initial data in all dimensions n 2 .

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in 3

M. Burak Erdoğan, Michael Goldberg, Wilhelm Schlag (2008)

Journal of the European Mathematical Society

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We present a novel approach for bounding the resolvent of H = - Δ + i ( A · + · A ) + V = : - Δ + L 1 for large energies. It is shown here that there exist a large integer m and a large number λ 0 so that relative to the usual weighted L 2 -norm, ( L ( - Δ + ( λ + i 0 ) ) - 1 ) m < 1 2 2 for all λ > λ 0 . This requires suitable decay and smoothness conditions on A , V . The estimate (2) is trivial when A = 0 , but difficult for large A since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

Unconditional uniqueness of higher order nonlinear Schrödinger equations

Friedrich Klaus, Peer Kunstmann, Nikolaos Pattakos (2021)

Czechoslovak Mathematical Journal

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We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data u 0 X , where X { M 2 , q s ( ) , H σ ( 𝕋 ) , H s 1 ( ) + H s 2 ( 𝕋 ) } and q [ 1 , 2 ] , s 0 , or σ 0 , or s 2 s 1 0 . Moreover, if M 2 , q s ( ) L 3 ( ) , or if σ 1 6 , or if s 1 1 6 and s 2 > 1 2 we show that the Cauchy problem is unconditionally wellposed in X . Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ...

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 .

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 - Δ ( a 11 u ) = h ( t , x ) | v | p , v - Δ ( a 21 u ) - Δ ( a 22 v ) = k ( t , x ) | w | q , w - Δ ( a 31 u ) - Δ ( a 32 v ) - Δ ( a 33 w ) = l ( t , x ) | u | r , for x 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 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...

On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations

Malkhaz Ashordia (2016)

Mathematica Bohemica

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The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense d x ( t ) = d A 0 ( t ) · x ( t ) + d f 0 ( t ) , x ( t 0 ) = c 0 ( t I ) with a unique solution x 0 is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems d x ( t ) = d A k ( t ) · x ( t ) + d f k ( t ) , x ( t k ) = c k ( k = 1 , 2 , ) to have a unique solution x k for any sufficiently large k such that x k ( t ) x 0 ( t ) uniformly on I . Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems....

Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

Andrea R. Nahmod, Gigliola Staffilani (2015)

Journal of the European Mathematical Society

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We also prove a long time existence result; more precisely we prove that for fixed T > 0 there exists a set Σ T , ( Σ T ) > 0 such that any data φ ω ( x ) H γ ( 𝕋 3 ) , γ < 1 , ω Σ T , evolves up to time T into a solution u ( t ) with u ( t ) - e i t Δ φ ω C ( [ 0 , T ] ; H s ( 𝕋 3 ) ) , s = s ( γ ) > 1 . In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space H 1 ( 𝕋 3 ) , that is in the supercritical scaling regime.

Logarithmically improved blow-up criterion for smooth solutions to the Leray- α -magnetohydrodynamic equations

Ines Ben Omrane, Sadek Gala, Jae-Myoung Kim, Maria Alessandra Ragusa (2019)

Archivum Mathematicum

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In this paper, the Cauchy problem for the 3 D Leray- α -MHD model is investigated. We obtain the logarithmically improved blow-up criterion of smooth solutions for the Leray- α -MHD model in terms of the magnetic field B only in the framework of homogeneous Besov space with negative index.

On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

Xuwen Chen, Justin Holmer (2016)

Journal of the European Mathematical Society

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We consider the 3D quantum BBGKY hierarchy which corresponds to the N -particle Schrödinger equation. We assume the pair interaction is N 3 β 1 V ( B β ) . For the interaction parameter β ( 0 , 2 / 3 ) , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the N limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for β ( 0 , 3 / 5 ) in [28]. This allows, in the case β ( 0 , 3 / 5 ) , for the application of the Klainerman–Machedon...

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.