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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

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

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: Shigehiro Sakata, Yuta Wakasugi. Movement of time-delayed hot spots in Euclidean space for a degenerate initial state. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2705-2738. doi: 10.3934/dcds.2020147
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.

[2]

W. Fulks and R. B. Guenther, Damped wave equations and the heat equation, Czechoslovak Math. J., 21 (1971), 683-695. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

A. Nikiforov and V. Ouvarov, \'Eléments de la Théorie des Fonctions Spéciales, \'Editions Mir, Moscow, 1976,256 pp.

[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.

[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.

[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.

show all references

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.

[2]

W. Fulks and R. B. Guenther, Damped wave equations and the heat equation, Czechoslovak Math. J., 21 (1971), 683-695. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

A. Nikiforov and V. Ouvarov, \'Eléments de la Théorie des Fonctions Spéciales, \'Editions Mir, Moscow, 1976,256 pp.

[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.

[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.

[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.

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 $
Table Options

Download as excel

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 $
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 $t\to \infty$
$\mathcal{M} (- u(\cdot ,t) )$ $\subset ( CS(f+g) + \varphi (t) B^n )^c$
Table Options

Download as excel

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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