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Klein-Gordon type decay rates for wave equations with time-dependent coefficients

Michael Reissig, Karen Yagdjian (2000)

Banach Center Publications

This work is concerned with the proof of L p - L q decay estimates for solutions of the Cauchy problem for the Klein-Gordon type equation u t t - λ 2 ( t ) b 2 ( t ) ( Δ u - m 2 u ) = 0 . The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. Moreover, m 2 is a positive constant. We study under which assumptions for λ and b one can expect as an essential part of the decay rate the classical Klein-Gordon decay rate n/2(1/p-1/q).

Korn's First Inequality with variable coefficients and its generalization

Waldemar Pompe (2003)

Commentationes Mathematicae Universitatis Carolinae

If Ω n is a bounded domain with Lipschitz boundary Ω and Γ is an open subset of Ω , we prove that the following inequality Ω | A ( x ) u ( x ) | p d x 1 / p + Γ | u ( x ) | p d n - 1 ( x ) 1 / p c u W 1 , p ( Ω ) holds for all u W 1 , p ( Ω ; m ) and 1 < p < , where ( A ( x ) u ( x ) ) k = i = 1 m j = 1 n a k i j ( x ) u i x j ( x ) ( k = 1 , 2 , ... , r ; r m ) defines an elliptic differential operator of first order with continuous coefficients on Ω ¯ . As a special case we obtain Ω u ( x ) F ( x ) + ( u ( x ) F ( x ) ) T p d x c Ω | u ( x ) | p d x , ( * ) for all u W 1 , p ( Ω ; n ) vanishing on Γ , where F : Ω ¯ M n × n ( ) is a continuous mapping with det F ( x ) μ > 0 . Next we show that ( * ) is not valid if n 3 , F L ( Ω ) and det F ( x ) = 1 , but does hold if p = 2 , Γ = Ω and F ( x ) is symmetric and positive definite in Ω .

L 2 -stability of multi-solitons

Claudio Muñoz (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

The aim of this note is to give a short review of our recent work (see [5]) with Miguel A. Alejo and Luis Vega, concerning the L 2 -stability, and asymptotic stability, of the N -soliton of the Korteweg-de Vries (KdV) equation.

L 2 -stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s

Salvatore Rionero (2005)

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

The L 2 -stability (instability) of a binary nonlinear reaction diffusion system of P.D.E.s - either under Dirichlet or Neumann boundary data - is considered. Conditions allowing the reduction to a stability (instability) problem for a linear binary system of O.D.E.s are furnished. A peculiar Liapunov functional V linked (together with the time derivative along the solutions) by direct simple relations to the eigenvalues, is used.

L -estimate for solutions of nonlinear parabolic systems

Wojciech Zajączkowski (1996)

Banach Center Publications

We prove existence of weak solutions to nonlinear parabolic systems with p-Laplacians terms in the principal part. Next, in the case of diagonal systems an L -estimate for weak solutions is shown under additional restrictive growth conditions. Finally, L -estimates for weakly nondiagonal systems (where nondiagonal elements are absorbed by diagonal ones) are proved. The L -estimates are obtained by the Di Benedetto methods.

L estimates of solution for m -Laplacian parabolic equation with a nonlocal term

Pulun Hou, Caisheng Chen (2011)

Czechoslovak Mathematical Journal

In this paper, we consider the global existence, uniqueness and L estimates of weak solutions to quasilinear parabolic equation of m -Laplacian type u t - div ( | u | m - 2 u ) = u | u | β - 1 Ω | u | α d x in Ω × ( 0 , ) with zero Dirichlet boundary condition in Ω . Further, we obtain the L estimate of the solution u ( t ) and u ( t ) for t > 0 with the initial data u 0 L q ( Ω ) ( q > ...

L p -decay of solutions to dissipative-dispersive perturbations of conservation laws

Grzegorz Karch (1997)

Annales Polonici Mathematici

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.

L p -Estimates for linear elliptic systems with discontinuous coefficients

Filippo Chiarenza, Michelangelo Franciosi, Michele Frasca (1994)

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

In this Note we give L p estimates 1 < p < + for the highest order derivatives of an elliptic system in non-divergence form with coefficients in VMO.

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