Time decay estimates for solutions of the Cauchy problem for the modified Kawahara equation

2019 ◽  
Vol 210 (5) ◽  
pp. 693-730
Author(s):  
P. I. Naumkin
Keyword(s):  
Cauchy Problem ◽  
Decay Estimates ◽  
Time Decay ◽  
Kawahara Equation ◽  
The Cauchy Problem

LANDAU–GINZBURG TYPE EQUATIONS IN THE SUBCRITICAL CASE

2003 ◽  
Vol 05 (01) ◽  
pp. 127-145 ◽  
Author(s):  
NAKAO HAYASHI ◽  
ELENA I. KAIKINA ◽  
PAVEL I. NAUMKIN
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Decay Rate ◽  
Unique Solution ◽  
Linear Case ◽  
Decay Estimates ◽  
Time Decay ◽  
Subcritical Case ◽  
Small Norm ◽  
The Cauchy Problem

We study the Cauchy problem for the nonlinear Landau–Ginzburg equation [Formula: see text] where α, β ∈ C with dissipation condition ℜα > 0. We are interested in the subcritical case [Formula: see text]. We assume that θ = | ∫ u0(x) dx| ≠ 0 and ℜδ (α, β) > 0, where [Formula: see text] Furthermore we suppose that the initial data u0 ∈ L1 are such that (1+|x|)au0 ∈ L1, with sufficiently small norm ε = ‖(1 + |x|)a u0 ‖1, where a ∈ (0,1). Also we assume that σ is sufficiently close to [Formula: see text]. Then there exists a unique solution of the Cauchy problem (*) such that [Formula: see text] satisfying the following time decay estimates for large t > 0[Formula: see text] Note that in comparison with the corresponding linear case the decay rate of the solutions of (*) is more rapid.

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

scholarly journals Time decay for solutions to the Stokes equations with drift

2018 ◽  
Vol 20 (03) ◽  
pp. 1750046 ◽  
Author(s):  
M. Schonbek ◽  
G. Seregin
Keyword(s):  
Cauchy Problem ◽  
Stokes System ◽  
Stokes Equations ◽  
Time Decay ◽  
Smooth Vector ◽  
Scale Invariant ◽  
The Cauchy Problem ◽  
Divergence Free ◽  
Vector Valued Function ◽  
Vector Valued

In this note, we study the behavior of Lebesgue norms [Formula: see text] of solutions [Formula: see text] to the Cauchy problem for the Stokes system with drift [Formula: see text], which is supposed to be a divergence free smooth vector-valued function satisfying a scale invariant condition.

Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝ N \mathbb{R}^{N}

2019 ◽  
Vol 31 (3) ◽  
pp. 803-814
Author(s):  
Ning Duan ◽  
Xiaopeng Zhao
Keyword(s):  
Cauchy Problem ◽  
Decay Rate ◽  
Weak Solutions ◽  
Well Posedness ◽  
Decay Estimates ◽  
Hilliard Equation ◽  
Temporal Decay ◽  
The Cauchy Problem ◽  
Higher Order Derivative ◽  
Cahn Hilliard Equation

AbstractThis paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in{\mathbb{R}^{N}}({N\in\mathbb{N}^{+}}), provided that the initial datum is sufficiently small. In addition, the{L^{p}}-norm ({1\leq p\leq\infty}) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied.

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