Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system

Takafumi Hosono, Shuichi Kawashima

Research output: Contribution to journalArticle

59 Citations (Scopus)

Abstract

We discuss the global solvability and asymptotic behavior of solutions to the Cauchy problem for some nonlinear hyperbolic-elliptic system with a fourth-order elliptic part. This system is a modified version of the simplest radiating gas model and verifies a decay property of regularity-loss type. Such a dissipative structure also appears in the dissipative Timoshenko system studied by Rivera and Racke. This dissipative property is very weak in high frequency region and causes the difficulty in deriving the desired a priori estimates for global solutions to the nonlinear problem. In fact, it turns out that the usual energy method does not work well. We overcome this difficulty by employing a time-weighted energy method which is combined with the optimal decay for lower order derivatives of solutions, and we establish a global existence and asymptotic decay result. Furthermore, we show that the solution has an asymptotic self-similar profile described by the Burgers equation as time tends to infinity.

Original languageEnglish
Pages (from-to)1839-1859
Number of pages21
JournalMathematical Models and Methods in Applied Sciences
Volume16
Issue number11
DOIs
Publication statusPublished - 2006 Jan 1
Externally publishedYes

Fingerprint

Elliptic Systems
Hyperbolic Systems
Regularity
Energy Method
Decay
Dissipative Structure
Global Solvability
Asymptotic Behavior of Solutions
Burgers Equation
A Priori Estimates
Global Solution
Global Existence
Fourth Order
Nonlinear Problem
Cauchy Problem
Infinity
Tend
Verify
Derivatives
Derivative

Keywords

  • Dissipative structure
  • Global solvability
  • Regularity-loss

ASJC Scopus subject areas

  • Modelling and Simulation
  • Applied Mathematics

Cite this

Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system. / Hosono, Takafumi; Kawashima, Shuichi.

In: Mathematical Models and Methods in Applied Sciences, Vol. 16, No. 11, 01.01.2006, p. 1839-1859.

Research output: Contribution to journalArticle

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