Improved spatial decay bounds in generalized heat conduction

Payne, L. E.; Song, J.C.

doi:10.1007/s00033-005-4010-x

Cited by 8 publications

(5 citation statements)

References 18 publications

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“…From a mechanical point of view, it is related to the Saint-Venant principle and, from a mathematical point of view, with the Phragmén-Lindelöf principle. Mathematical studies about the spatial behavior have been proposed for elliptic, hyperbolic and parabolic equations [2,[5][6][7][8][11][12][13][14]19,20,[23][24][25]. The list of contributions in this theory is huge, but we want to focus our attention to the parabolic case.…”

Section: Introductionmentioning

confidence: 99%

Fast spatial behavior in higher order in time equations and systems

Fernández

Quintanilla

2022

Z. Angew. Math. Phys.

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In this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The thermoelastic case is also considered when the heat conduction is modeled using a high-order parabolic equation. Though the arguments are similar to others usually applied, we obtain new relevant results by selecting appropriate functions never considered before.

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Section: Introductionmentioning

confidence: 99%

Fast spatial behavior in higher order in time equations and systems

Fernández

Quintanilla

2022

Z. Angew. Math. Phys.

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“…From a mathematical perspective, the spatial behavior of the solutions is an important issue to be studied. [37][38][39][40][41][42] It is worth noting that the uniqueness of solutions cannot be expected in this context. In fact, the problem is ill posed in the sense of Hadamard, and only a Phragmén-Lindelöf alternative for the solutions can be obtained.…”

Section: Introductionmentioning

confidence: 99%

Spatial behavior in high‐order partial differential equations

Leseduarte

Quintanilla

2018

Math Methods in App Sciences

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In this paper, we study the spatial behavior of solutions to the equations obtained by taking formal Taylor approximations to the heat conduction dual‐phase‐lag and 3‐phase‐lag theories, reflecting Saint‐Venant's principle. In a recent paper, 2 families of cases for high‐order partial differential equations were studied. Here, we investigate a third family of cases, which corresponds to the fact that a certain condition on the time derivative must be satisfied. We also study the spatial behavior of a thermoelastic problem. We obtain a Phragmén‐Lindelöf alternative for the solutions in both cases. The main tool to handle these problems is the use of an exponentially weighted Poincaré inequality.

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“…Later it was shown by Knowles [11] that in the case of classical heat conduction in a finite www cylinder with appropriate homogeneous boundary and initial conditions applied except on one of the cylinders, then the temperature in energy measure decays away from that end at least as fast as that of its elliptic counterpart. Saint-Venant type decay bounds for time dependent parabolic equations may be found in [4,5,[15][16][17][18]22,23,25]. To establish spatial decay estimates of Saint-Venant type, one always has to assume that solution must satisfy some a priori decay criterions at infinity.…”

Section: Introductionmentioning

confidence: 99%