Global in time existence of small solutions of non‐linear thermoviscoelastic equations

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12 Citations (Scopus)

Abstract

We prove a global in time existence theorem of classical solutions of the initial boundary value problem for a non‐linear thermoviscoelastic equation in a bounded domain for very smooth initial data, external forces and heat supply which are very close to a specific constant equilibrium state. Our proof is a combination of a local in time existence theorem and some a priori estimates of local in time solutions. Such a priori estimates are proved basically for suitable linear problems by using some multiplicative techniques. An exponential stability of the constant equilibrium state also follows from our proof of the existence and regularity theorems.

Original languageEnglish
Pages (from-to)871-895
Number of pages25
JournalMathematical Methods in the Applied Sciences
Volume18
Issue number11
DOIs
Publication statusPublished - 1995
Externally publishedYes

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Small Solutions
Equilibrium constants
Nonlinear equations
Nonlinear Equations
A Priori Estimates
Equilibrium State
Existence Theorem
Asymptotic stability
Boundary value problems
Exponential Stability
Classical Solution
Initial-boundary-value Problem
Bounded Domain
Multiplicative
Heat
Regularity
Theorem
Hot Temperature

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)

Cite this

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title = "Global in time existence of small solutions of non‐linear thermoviscoelastic equations",
abstract = "We prove a global in time existence theorem of classical solutions of the initial boundary value problem for a non‐linear thermoviscoelastic equation in a bounded domain for very smooth initial data, external forces and heat supply which are very close to a specific constant equilibrium state. Our proof is a combination of a local in time existence theorem and some a priori estimates of local in time solutions. Such a priori estimates are proved basically for suitable linear problems by using some multiplicative techniques. An exponential stability of the constant equilibrium state also follows from our proof of the existence and regularity theorems.",
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AB - We prove a global in time existence theorem of classical solutions of the initial boundary value problem for a non‐linear thermoviscoelastic equation in a bounded domain for very smooth initial data, external forces and heat supply which are very close to a specific constant equilibrium state. Our proof is a combination of a local in time existence theorem and some a priori estimates of local in time solutions. Such a priori estimates are proved basically for suitable linear problems by using some multiplicative techniques. An exponential stability of the constant equilibrium state also follows from our proof of the existence and regularity theorems.

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