Growth estimates for solutions of evolutionary equations in Hilbert space with applications in elastodynamics

Knops, R. J.; Payne, L. E.

doi:10.1007/bf00281873

Cited by 73 publications

(61 citation statements)

References 9 publications

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“…The case (b) follows immediately by taking ω = 0. Finally, case (c) follows the calculations of Knops and Payne ( [13], see equation (3.6.14)) and the theorem is proved. …”

Section: In This Situation An Energy Equation Ismentioning

confidence: 65%

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Some remarks on growth and uniqueness in thermoelasticity

Quintanilla

2003

International Journal of Mathematics and Mathematical Sciences

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We use the Lagrange identity method and the logarithmic convexity to obtain uniqueness and exponential growth of solutions in the thermoelasticity of type III and thermoelasticity without energy dissipation. As this is not the first contribution of this kind in this theory, it is worth remarking that the assumptions we use here are different from those used in other previous contributions. We assume that the elasticity tensor is positive semidefinite, but we allow that the constitutive tensor of the entropy flux vector (k ij ), which is a characteristic tensor in this theory, is not sign-definite. The Lagrange identity method is used to obtain uniqueness in the context of the thermoelasticity of type III. The fundamental key to obtain exponential growth in the thermoelasticity without energy dissipation is the use of a new functional. This functional is inspired in that it is used when the elasticity tensor is not sign-definite, but (k ij ) is positive definite.

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“…The case (b) follows immediately by taking ω = 0. Finally, case (c) follows the calculations of Knops and Payne ( [13], see equation (3.6.14)) and the theorem is proved. …”

Section: In This Situation An Energy Equation Ismentioning

confidence: 65%

“…The addition of the term ω(t +t 0 ) 2 is an idea of Knops and Payne [13] in the context of isothermal elasticity. By differentiation, we see that…”

Section: In This Situation An Energy Equation Ismentioning

confidence: 99%

Some remarks on growth and uniqueness in thermoelasticity

Quintanilla

2003

International Journal of Mathematics and Mathematical Sciences

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“…Murray [11] considered ultrahyperbolic operators, Ogawa [14] discussed hyperbolic operators, Knops and Payne [3,4], and Knops, Levine and Payne [2] investigated both the abstract problem and its applications to elasticity.…”

Section: G N Hile and M H Prottermentioning

confidence: 99%

The Cauchy problem and asymptotic decay for solutions of differential inequalities in Hilbert space

Hile¹,

Protter²

1982

Pacific J. Math.

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“…The technique used in [ 1 ] - [3] is based on a logarithmic convexity argument first employed by Knops and Payne [5] for the abstract wave equation obtained from ( 1. 1 ) by setting K(t) 0; a different logarithmic convexity argument was employed by this author in [ 4J to derive continuous data dependence theorems for the system ( 1.…”

mentioning

confidence: 99%

Bounds for Solutions to a Class of Integro-differential Equations Associated with a Theory of Rigid Nonconducting Material Dielectrics

Bloom¹

1980

SIAM J. Math. Anal.

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Growth estimates for solutions of evolutionary equations in Hilbert space with applications in elastodynamics

Cited by 73 publications

References 9 publications

Some remarks on growth and uniqueness in thermoelasticity

Some remarks on growth and uniqueness in thermoelasticity

The Cauchy problem and asymptotic decay for solutions of differential inequalities in Hilbert space

Bounds for Solutions to a Class of Integro-differential Equations Associated with a Theory of Rigid Nonconducting Material Dielectrics

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