# American Institute of Mathematical Sciences

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 - A, 2020, 40 (5) : 2705-2738. doi: 10.3934/dcds.2020147
##### References:
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##### References:
 [1] I. Chavel and L. Karp, Movement of hot spots in Riemannian manifolds, J. Analyse Math., 55 (1990), 271-286.  doi: 10.1007/BF02789205.  Google Scholar [2] W. Fulks and R. B. Guenther, Damped wave equations and the heat equation, Czechoslovak Math. J., 21 (1971), 683-695.   Google Scholar [3] T. Hosono and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118.  doi: 10.1016/j.jde.2004.03.034.  Google Scholar [4] R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain, J. Differential Equations, 186 (2002), 633-651.  doi: 10.1016/S0022-0396(02)00008-6.  Google Scholar [5] S. Jimbo and S. Sakaguchi, Movement of hot spots over unbounded domains in $\mathbb{R}^N$, J. Math. Anal. Appl., 182 (1994), 810-835.  doi: 10.1006/jmaa.1994.1123.  Google Scholar [6] T.-T. Li, Nonlinear heat conduction with finite speed of propagation, China-Japan Symposium on Reaction-Diffusion Equations and Their Applications and Computational Aspects (Shanghai, 1994), World Sci. Publ., River Edge, NJ, (1997), 81–91.  Google Scholar [7] P. Marcati and K. Nishihara, The $L^p$-$L^q$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445-469.  doi: 10.1016/S0022-0396(03)00026-3.  Google Scholar [8] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976/77), 169-189.  doi: 10.2977/prims/1195190962.  Google Scholar [9] T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626.  doi: 10.2969/jmsj/1191418647.  Google Scholar [10] A. Nikiforov and V. Ouvarov, \'Eléments de la Théorie des Fonctions Spéciales, \'Editions Mir, Moscow, 1976,256 pp.  Google Scholar [11] K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649.  doi: 10.1007/s00209-003-0516-0.  Google Scholar [12] S. Sakata and Y. Wakasugi, Movement of time-delayed hot spots in Euclidean space, Math. Z., 285 (2017), 1007-1040.  doi: 10.1007/s00209-016-1735-5.  Google Scholar [13] H. Yang and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves, Bull. Sci. Math., 124 (2000), 415-433.  doi: 10.1016/S0007-4497(00)00141-X.  Google Scholar
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[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 $t\to \infty$ $\mathcal{M} (- u(\cdot ,t) )$ $\subset ( CS(f+g) + \varphi (t) B^n )^c$
 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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