Movement of time-delayed hot spots in Euclidean space for a degenerate initial state

Shigehiro Sakata; Yuta Wakasugi

doi:10.3934/dcds.2020147

Article Contents

2020, Volume 40, Issue 5: 2705-2738. Doi: 10.3934/dcds.2020147

This issue Previous Article A variational principle of topological pressure on subsets for amenable group actions Next Article Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement

Movement of time-delayed hot spots in Euclidean space for a degenerate initial state

Shigehiro Sakata^1, and
Yuta Wakasugi^2,

1.
Department of Applied Mathematics, Fukuoka University, 8-19-1 Nanakuma, Jonan, Fukuoka, 814-0180, Japan
2.
Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

Received: May 2019

Revised: December 2019

Published: March 2020

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Abstract

We consider the Cauchy problem for the damped wave equation under the initial state that the sum of an initial position and an initial velocity vanishes. When the initial position is non-zero, non-negative and compactly supported, we study the large time behavior of the spatial null, critical, maximum and minimum sets of the solution. The behavior of each set is totally different from that of the corresponding set under the initial state that the sum of an initial position and an initial velocity is non-zero and non-negative.

The spatial null set includes a smooth hypersurface homeomorphic to a sphere after a large enough time. The spatial critical set has at least three points after a large enough time. The set of spatial maximum points escapes from the convex hull of the support of the initial position. The set of spatial minimum points consists of one point after a large enough time, and the unique spatial minimum point converges to the centroid of the initial position at time infinity.

Keywords:

Damped wave equation,
heat equation with finite propagation speed,
diffusion phenomenon,
time-delayed hot spots,
Nishihara decomposition.

Mathematics Subject Classification: Primary: 35L15, 35B38; Secondary: 35C15, 35B40, 35K05.

Citation:

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Table 1. Results of this paper (when $ f+g = 0 $)

The object	Its property
$ \mathcal{N} ( u(\cdot ,t) ) \cap ( CS(f)+ \varphi (t )B^n ) $	$ \subset A_f^\circ ( \sqrt{2nt}-d_f , \sqrt{2nt} ) $, $ \approx S^{n-1} $
$ \mathcal{C} ( u(\cdot ,t) ) \cap ( CS(f)+ \varphi (t )B^n ) $	$ \subset A_f^\circ ( \sqrt{(2n+4)t}-d_f , \sqrt{(2n+4)t} ) \cup CS(f), \sharp \geq 3$
$ \mathcal{M} ( u(\cdot ,t) ) $	$ \subset A_f^\circ ( \sqrt{(2n+4)t}-d_f , \sqrt{(2n+4)t} ) $
$ \mathcal{M} (- u(\cdot ,t) ) $	$ \subset CS(f) $, $ \sharp =1 $, $ \to \left\{ {m_f} \right\} $ as $ t \to \infty $

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Table 2. Results of[12](when $f+g = 0$)

The object	Its property
$\mathcal{N} ( u(\cdot ,t) ) \cap ( CS(f+g)+ \varphi (t )B^n )$	$\subset (CS (f+g) + \varphi (t) B^n )^c$
$\mathcal{C} ( u(\cdot ,t) ) \cap ( CS(f+g)+ \varphi (t )B^n )$	$\subset CS (f+g)$, $\sharp = 1$
$\mathcal{M} ( u(\cdot ,t) )$	$\subset CS (f+g)$, $\sharp =1$, $\to \{ m_{f+g} \}$ as $t\to \infty$
$\mathcal{M} (- u(\cdot ,t) )$	$\subset ( CS(f+g) + \varphi (t) B^n )^c$

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