# 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:
 [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

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.  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 $t\to \infty$ $\mathcal{M} (- u(\cdot ,t) )$ $\subset ( CS(f+g) + \varphi (t) B^n )^c$
 [1] Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $\mathbb{R}^n_+$. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033 [2] Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270 [3] Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 [4] Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 [5] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [6] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [7] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [8] Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175 [9] Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004 [10] Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021008 [11] Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015 [12] Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047 [13] Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365 [14] Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388 [15] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [16] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [17] Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364 [18] Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034 [19] Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 [20] Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160

2019 Impact Factor: 1.338