Periodic problem for the nonlinear damped wave equation with convective nonlinearity

Carreño‐Bolaños, Rafael; Juárez-Campos, Beatriz; Naumkin, Pavel I.

doi:10.1111/sapm.12316

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…We represent 𝑦 = 𝑟𝑒 𝑖𝜑 , then from Equation ( 20) we get 𝑟 ′ + 𝑟𝑖𝜑 ′ = −𝑎𝑟 3 − 𝑖𝑏𝑟 3 + 𝑅𝑒 −𝑖𝜑 . Hence taking real and imaginary parts we find 𝑟 ′ = −(𝑎 − 𝑄 1 )𝑟 3 and 𝜑 ′ = −𝑏𝑟 2 + 𝑄 2 , where 𝑄 1 = Re 𝑟 −3 𝑅𝑒 −𝑖𝜑 , 𝑄 2 = Im 𝑟 −1 𝑅𝑒 −𝑖𝜑 . Integration with respect to time yields…”

Section: Preliminary Estimatesmentioning

confidence: 98%

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Asymptotics of solutions to periodic problem for the Korteweg–de Vries–Burgers equation

Naumkin

Villela-Aguilar

2022

Stud Appl Math

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We consider the periodic problem for the Korteweg–de Vries–Burgers (KdVB) equation with pumping vtgoodbreak−vxxgoodbreak+α3vxxxgoodbreak=vgoodbreak+∂x()v2,x∈normalΩ,tgoodbreak>0,v(0,x)goodbreak=v0()x,x∈normalΩ,\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{c}v_{t}-v_{xx}+\frac{\alpha }{3}v_{xxx}=v+\partial _{x}{\left(v^{2}\right)} ,\text{ }x\in \Omega ,t>0,\\ v(0,x)=v_{0}{\left(x\right)} ,\text{ }x\in \Omega , \end{array} \right.} \end{equation*}where normalΩ=false[−π,πfalse]$\Omega =[ -\pi ,\pi ]$, α∈double-struckR$\alpha \in \mathbb {R}$. We study the solutions, which satisfy the periodic boundary conditions vfalse(t,xfalse)=vfalse(t,2π+xfalse)$v(t,x) =v(t,2\pi +x)$ for all x∈double-struckR$x\in \mathbb {R}$ and t>0$t>0$, with the 2π—periodic initial data v0(x)$v_{0}(x)$. Our aim is to find large time asymptotic profile of solutions. The large time asymptotic behavior of solutions to the Cauchy problem for the KdVB equations with different dissipative terms was extensively studied. In the present paper we find the asymptotic profile of solutions for large time. We develop the approach started in Refs. [1, 2, 3]. We prove the following asymptotics for the solutions: vt,x=met+ε1+2aε2tcosnormalΛ−b2alog()1+2aε2t+O1+ε2t−1\begin{align*} & v{\left(t,x\right)} =me^{t}\\ & +\frac{\varepsilon }{\sqrt {1+2a\varepsilon ^{2}t}}\cos {\left(\Lambda -\frac{b}{2a}\log {\left(1+2a\varepsilon ^{2}t\right)} \right)} \\ & +O{\left({\left(1+\varepsilon ^{2}t\right)} ^{-1}\right)} \end{align*}as t→∞$t\rightarrow \infty$ uniformly with respect to x∈normalΩ$x\in \Omega$, where m=∫Ωv0(x)dx$m=\int _{\Omega }v_{0}(x) dx$, ε=|∫Ωeixv0false(xfalse)dx|$\varepsilon =|\int _{\Omega }e^{ix}v_{0}(x) dx|$, a=12|μ|2$a=\frac{12}{|\mu |^{2}}$, b=8α|μ|2$b=\frac{8\alpha }{|\mu |^{2}}$, μ=3−2iα$\mu =3-2i\alpha$, Λ is a constant.

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Section: Preliminary Estimatesmentioning

confidence: 98%

“…We develop the approach started in Refs. [1,2,3]. We prove the following asymptotics for the solutions:…”

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confidence: 90%

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Asymptotics of solutions to periodic problem for the Korteweg–de Vries–Burgers equation

Naumkin

Villela-Aguilar

2022

Stud Appl Math

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Asymptotics of Solutions for Periodic Problem for the Korteweg-de Vries Equation with Landau Damping, Pumping and Higher Order Convective Non Linearity

Juárez-Campos,

Villela-Aguilar,

Carreño-Bolaños

2023

J Nonlinear Math Phys

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We study the periodic problem for the Korteweg–de Vries equation with Landau damping, linear pumping and a higher-order convective nonlinearity $$\begin{aligned} \left\{ \begin{array}{c} w_{t}+w_{xxx}-\alpha w_{xx}=\beta w+\lambda w_{x}^{2}w_{xx},\text { }x\in \Omega ,t>0,\\ w(0,x)=\psi \left( x\right) ,\text { }x\in \Omega , \end{array} \right. \end{aligned}$$ w t + w xxx - α w xx = β w + λ w x 2 w xx , x ∈ Ω , t > 0 , w ( 0 , x ) = ψ x , x ∈ Ω , where, $$\alpha ,\beta >0,$$ α , β > 0 , $$\lambda \in \mathbb {R},$$ λ ∈ R , $$\Omega =\left[ -\pi ,\pi \right] .$$ Ω = - π , π . We assume that the initial data $$\psi \left( x\right) $$ ψ x are $$2\pi $$ 2 π - periodic. We prove the global existence of solutions and analyze their large-time asymptotics.

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Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation with convective nonlinearity

Carreño-Bolaños,

Naumkin

2024

Nonlinear Analysis

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Periodic problem for the nonlinear damped wave equation with convective nonlinearity

Cited by 3 publications

References 9 publications

Asymptotics of solutions to periodic problem for the Korteweg–de Vries–Burgers equation

Asymptotics of solutions to periodic problem for the Korteweg–de Vries–Burgers equation

Asymptotics of Solutions for Periodic Problem for the Korteweg-de Vries Equation with Landau Damping, Pumping and Higher Order Convective Non Linearity

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation with convective nonlinearity

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