Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

Mohamed Ali Hamza, Yuta Wakasugi, Shuji Yoshikawa

TL;DR

This work analyzes the Cauchy problem for a nonlinear damped beam equation with two time-dependent coefficients, aiming to identify conditions under which solutions exhibit heat-like asymptotics. Using a self-similar scaling and a Gaussian-remnant decomposition, the authors build a comprehensive energy framework with a hierarchy of Lyapunov functionals to control the nonlinear remainder; Hardy-type inequalities and careful tracking of the variable-coefficient damping ensure the nonlinear terms act as small remainders. Under Assumptions (A) and (N) with $(\alpha,\beta)\in\Omega_1$ and small initial data, they prove global existence and establish that the solution converges to a heat-kernel profile $u(t,\cdot)\to m^{*} G(R(t),\cdot)$, with $R(t)=\int_0^t a(\tau)/b(\tau)\,d\tau$ and rate $\|u(t,\cdot)-m^{*}G(R(t),\cdot)\|_{L^{2}} \lesssim (R(t)+1)^{-1/4-\lambda/2}$ for $0<\lambda<\min\{1/2, 2(\beta+1)/(\alpha-\beta+1), (2\alpha-\beta+1)/(\alpha-\beta+1)\}$. This extends linear-damping results to a nonlinear setting, showing that effective damping drives the long-time behavior toward a diffusion-dominated profile despite nonlinear derivative terms. The analysis combines a Gallay–Raugel-style self-similar transform, a detailed energy-method hierarchy, and careful remainder estimates to obtain precise decay and asymptotic profiles.

Abstract

In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients: \[ \partial_t^2 u + b(t) \partial_t u - a(t) \partial_x^2 u + \partial_x^4 u = \partial_x \left( N(\partial_x u) \right). \] In the authors' previous article [17], the asymptotic profile of solutions for linearized problem ($N \equiv 0$) was classified depending on the assumptions for the coefficients $a(t)$ and $b(t)$ and proved the asymptotic behavior in effective damping cases. We here give the conditions of the coefficients and the nonlinear term in order that the solution behaves as the solution for the heat equation: $b(t) \partial_t u - a(t) \partial_x^2 u=0$ asymptotically as $t \to \infty$.

Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

TL;DR

This work analyzes the Cauchy problem for a nonlinear damped beam equation with two time-dependent coefficients, aiming to identify conditions under which solutions exhibit heat-like asymptotics. Using a self-similar scaling and a Gaussian-remnant decomposition, the authors build a comprehensive energy framework with a hierarchy of Lyapunov functionals to control the nonlinear remainder; Hardy-type inequalities and careful tracking of the variable-coefficient damping ensure the nonlinear terms act as small remainders. Under Assumptions (A) and (N) with and small initial data, they prove global existence and establish that the solution converges to a heat-kernel profile , with and rate for . This extends linear-damping results to a nonlinear setting, showing that effective damping drives the long-time behavior toward a diffusion-dominated profile despite nonlinear derivative terms. The analysis combines a Gallay–Raugel-style self-similar transform, a detailed energy-method hierarchy, and careful remainder estimates to obtain precise decay and asymptotic profiles.

Abstract

In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients: In the authors' previous article [17], the asymptotic profile of solutions for linearized problem () was classified depending on the assumptions for the coefficients and and proved the asymptotic behavior in effective damping cases. We here give the conditions of the coefficients and the nonlinear term in order that the solution behaves as the solution for the heat equation: asymptotically as .
Paper Structure (9 sections, 16 theorems, 168 equations)

This paper contains 9 sections, 16 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Scaling variables
  3. Energy estimates
  4. Energy estimates
  5. Estimates of remainder terms and the proof of a priori estimate
  6. Estimates of remainder terms
  7. Proof of a priori estimate
  8. A general lemma for the energy identity
  9. Local existence

Key Result

Theorem 1.2

Under the assumptions (A) and (N), there exists a constant $\varepsilon_{0} > 0$ such that if $(u_{0}, u_{1}) \in \left( H^{2,1}(\mathbb{R}) \cap H^{3,0}(\mathbb{R}) \right) \times \left( H^{0,1}(\mathbb{R})\cap H^{1,0}(\mathbb{R}) \right)$ and then there exists a unique solution Moreover, the solution $u$ has the asymptotic behavior with some constants $C>0$, $m^{*} \in \mathbb{R}$, and $\lamb

Theorems & Definitions (30)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2: Hardy-type inequality Wa17
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 20 more