Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics

Knops, R. J.; Levine, Howard A.; Payne, L. E.

doi:10.1007/bf00282433

Cited by 73 publications

(46 citation statements)

References 9 publications

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“…See for example [2], [12], [13], [14], [15], [21], [22], [29], [34]. We also refer to the related papers [23] and [24], dealing with boundary source terms.…”

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confidence: 99%

A potential well theory for the wave equation with nonlinear source and boundary damping terms

Vitillaro

2002

Glasgow Math. J.

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Abstract. The paper deals with local existence, blow-up and global existence for the solutions of a wave equation with an internal nonlinear source and a nonlinear boundary damping. The typical problem studied is; 1Þ Â À 1 ; uð0; xÞ ¼ u 0 ðxÞ; u t ð0; xÞ ¼ u 1 ðxÞ on ;where & R n (n ! 1) is a regular and bounded domain,1 Þ, ! 0, and the initial data are in the energy space. The results proved extend the potential well theory, which is well known when the nonlinear damping acts in the interior of , to this problem.

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“…See for example [2], [12], [13], [14], [15], [21], [22], [29], [34]. We also refer to the related papers [23] and [24], dealing with boundary source terms.…”

mentioning

confidence: 99%

A potential well theory for the wave equation with nonlinear source and boundary damping terms

Vitillaro

2002

Glasgow Math. J.

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“…In this section, we use a 'concavity method' approach ( [6], [7]) to demonstrate finite time blowup of solutions to (27). Since solutions to the (equivalent) autonomous system (29), (30) define a local oneparameter group, (r, s)(t 2 + t 1 ; (r 0 , r 1 )) = (r, s)(t 2 ; ((r, s)(t 1 ; (r 0 , r 1 ))) where (r, s)(t; (r 0 , r 1 )) denotes a solution such that (r, s)(0; (r 0 , r 1 )) = (r 0 , r 1 ), this means that we may, for convenience of notation, continue to represent trajectories in U − (see (38)) as starting from t = 0.…”

Section: Singularity Formationmentioning

confidence: 99%

On the influence of damping in hyperbolic equations with parabolic degeneracy

Saxton

2011

Quart. Appl. Math.

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This paper examines the effect of damping on a nonstrictly hyperbolic 2×2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.

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“…Consequently, inequality (4.13) becomes 19) which after an application of the arithmetic-mean geometric mean inequality gives 20) where γ is an arbitrary positive constant.…”

Section: Definition 45 (Strict Rank-one Convexity) the Function W ∈mentioning

confidence: 99%

“…A similar result is established in [8] for a poly-convex entropy subject to certain other conditions. A different convexity assumption, further examined by Sofer [25], is adopted by Knops, Levine and Payne [20], who likewise establish uniqueness as a consequence of continuous dependence estimates.…”

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confidence: 99%

On local uniqueness in nonlinear elastodynamics

Knops

2006

Quart. Appl. Math.

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Abstract.A conservation law, derived from properties of the energy-momentum tensor, is used to establish uniqueness of suitably constrained solutions to the initial boundary value problem of nonlinear elastodynamics. It is assumed that the region is star-shaped, that the data are affine, and that the strain-energy function is strictly rank-one convex and quasi-convex. It is shown how these assumptions may be successively relaxed provided that the class of considered solutions is correspondingly further constrained.1. Introduction. This paper examines uniqueness of locally smooth solutions to the initial displacement boundary value problem of nonlinear homogeneous elastodynamics in the absence of body-forces and on a bounded three-dimensional region. Weak solutions are not necessarily unique, and smooth solutions do not in general exist globally in time. Consequently, uniqueness is sought only in the class of sufficiently smooth solutions on the maximal time interval of existence. s is the usual Hilbert space of order s, and s > n/2 + 1. The discussion in [7], that extends to thermoelastodynamics, considers a strongly elliptic entropy function and from estimates for continuous dependence on initial data obtains uniqueness of a smooth solution in the class of weak solutions. A similar result is established in [8] for a poly-convex entropy subject to certain other conditions. A different convexity assumption, further examined by Sofer [25], is adopted by Knops, Levine and Payne [20], who likewise establish uniqueness as a consequence of continuous dependence estimates.

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Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics

Cited by 73 publications

References 9 publications

A potential well theory for the wave equation with nonlinear source and boundary damping terms

A potential well theory for the wave equation with nonlinear source and boundary damping terms

On the influence of damping in hyperbolic equations with parabolic degeneracy

On local uniqueness in nonlinear elastodynamics

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