    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 and Continuous Dynamical Systems, 2020, 40 (5) : 2705-2738. doi: 10.3934/dcds.2020147
##### References:
  I. Chavel and L. Karp, Movement of hot spots in Riemannian manifolds, J. Analyse Math., 55 (1990), 271-286.  doi: 10.1007/BF02789205.   W. Fulks and R. B. Guenther, Damped wave equations and the heat equation, Czechoslovak Math. J., 21 (1971), 683-695.  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.   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.   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.   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.  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.   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.   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.   A. Nikiforov and V. Ouvarov, \'Eléments de la Théorie des Fonctions Spéciales, \'Editions Mir, Moscow, 1976,256 pp.  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.   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.   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:
  I. Chavel and L. Karp, Movement of hot spots in Riemannian manifolds, J. Analyse Math., 55 (1990), 271-286.  doi: 10.1007/BF02789205.   W. Fulks and R. B. Guenther, Damped wave equations and the heat equation, Czechoslovak Math. J., 21 (1971), 683-695.  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.   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.   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.   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.  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.   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.   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.   A. Nikiforov and V. Ouvarov, \'Eléments de la Théorie des Fonctions Spéciales, \'Editions Mir, Moscow, 1976,256 pp.  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.   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.   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.   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 $t\to \infty$ $\mathcal{M} (- u(\cdot ,t) )$ $\subset ( CS(f+g) + \varphi (t) B^n )^c$
  Kazuhiro Ishige, Y. Kabeya. Hot spots for the two dimensional heat equation with a rapidly decaying negative potential. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 833-849. doi: 10.3934/dcdss.2011.4.833  Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257  Lihua Min, Xiaoping Yang. Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 543-566. doi: 10.3934/cpaa.2014.13.543  Jean-Daniel Djida, Juan J. Nieto, Iván Area. Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4031-4053. doi: 10.3934/dcdsb.2019049  S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305  Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47  Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219  Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065  Yong Zhou, Zhengguang Guo. Blow up and propagation speed of solutions to the DGH equation. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 657-670. doi: 10.3934/dcdsb.2009.12.657  Julie Valein. On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021039  V. Pata, Sergey Zelik. A remark on the damped wave equation. Communications on Pure and Applied Analysis, 2006, 5 (3) : 611-616. doi: 10.3934/cpaa.2006.5.611  Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526  James B. Kennedy, Jonathan Rohleder. On the hot spots of quantum graphs. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3029-3063. doi: 10.3934/cpaa.2021095  Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure and Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41  David Henry. Infinite propagation speed for a two component Camassa-Holm equation. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 597-606. doi: 10.3934/dcdsb.2009.12.597  Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems and Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019  Antoine Benoit. Finite speed of propagation for mixed problems in the $WR$ class. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2351-2358. doi: 10.3934/cpaa.2014.13.2351  Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.  Chunpeng Wang, Jingxue Yin, Bibo Lu. Anti-shifting phenomenon of a convective nonlinear diffusion equation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1211-1236. doi: 10.3934/dcdsb.2010.14.1211  Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic and Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048

2020 Impact Factor: 1.392