    May  2020, 40(5): 2705-2738. doi: 10.3934/dcds.2020147

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

 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

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.

Citation: Shigehiro Sakata, Yuta Wakasugi. Movement of time-delayed hot spots in Euclidean space for a degenerate initial state. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2705-2738. doi: 10.3934/dcds.2020147
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##### References:
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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 $t \to \infty$
 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 $t \to \infty$
Results of(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 $t\to \infty$ $\mathcal{M} (- u(\cdot ,t) )$ $\subset ( CS(f+g) + \varphi (t) B^n )^c$
 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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