zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global existence and asymptotic behavior of solutions for a semi-linear wave equation. (English) Zbl 1232.35099
This work is concerned with the Cauchy initial value problem of a damped semilinear wave equation in n-dimensional Euclidean space. If we call the first time derivative of the dependent variable u as the velocity and its second time derivative as the acceleration, the linear part of the equation consists of the acceleration, the Laplacian of u, the Laplacian of the acceleration and a linear damping term given by the velocity. The non-linear damping is characterised by a smooth non-linear function of the velocity which is assumed to be of the order of the square of the velocity as the velocity tends to zero. Initial conditions for u and the velocity are prescribed functions at t=0. In order to attack the non-linear Cauchy problem, the authors first investigate the decay estimate of solutions of the linear damped wave equation by applying a Fourier transform on space variables and then study the solution of the resulting time-dependent ordinary differential equation. Based on this estimate, they define a set of time-weighted Sobolev spaces leading to a Banach space into which the solution functions are placed. Then, they prove the global existence and uniqueness of the solution of the semilinear equation by employing the contraction mapping theorem.
MSC:
35L71Semilinear second-order hyperbolic equations
35L05Wave equation (hyperbolic PDE)
35L20Second order hyperbolic equations, boundary value problems
References:
[1]Charão, R. C.; Bisognin, E.; Bisognin, V.; Pazoto, A. F.: Asymptotic behavior for a dissipative plate equation in RN with periodic coefficients, Electron. J. Differential equations 46, 1-23 (2008) · Zbl 1170.35319 · doi:emis:journals/EJDE/Volumes/2008/46/abstr.html
[2]Chen, X. -Y.; Chen, G. -W.: Asymptotic behavior and blow-up of solutions to a nonlinear evolution equation of fourth order, Nonlinear anal. 68, 892-904 (2008)
[3]Dai, W. -R.; Kong, D. -X.: Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields, J. differential equations 235, 127-165 (2007) · Zbl 1129.35047 · doi:10.1016/j.jde.2006.12.020
[4]Hörmander, L.: Lectures on nonlinear hyperbolic differential equations, Math. appl. (Springer) 26 (1997)
[5]Kong, D. -X.; Yang, T.: Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Comm. partial differential equations 28, 1203-1220 (2003) · Zbl 1024.35068 · doi:10.1081/PDE-120021192
[6]Liu, Z.; Zheng, S.: On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. appl. Math. 54, 21-31 (1996) · Zbl 0868.35011
[7]Liu, Z.; Zheng, S.: Semigroups associated with dissipative systems, (1999) · Zbl 0924.73003
[8]Liu, Y. -Q.; Wang, W. -K.: The pointwise estimates of solutions for dissipative wave equation in multi-dimensions, Discrete contin. Dyn. syst. 20, 1013-1028 (2008) · Zbl 1154.35014 · doi:10.3934/dcds.2008.20.1013
[9]Liu, Y. -Q.; Kawashima, S.: Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation, Discrete contin. Dyn. syst. 29, 1113-1139 (2011) · Zbl 1215.35035 · doi:10.3934/dcds.2011.29.1113
[10]Nakao, M.; Ono, K.: Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z. 214, 325-342 (1993) · Zbl 0790.35072 · doi:10.1007/BF02572407
[11]Nishihara, K.: Lp-lq estimates of solutions to the damped wave equation in 3-dimensional space and their applications, Math. Z. 244, 631-649 (2003) · Zbl 1023.35078 · doi:10.1007/s00209-003-0516-0
[12]Nishihara, K.; Zhao, H.: Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption, J. math. Anal. appl. 313, 598-610 (2006) · Zbl 1160.35479 · doi:10.1016/j.jmaa.2005.08.059
[13]Ono, K.: Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations, Discrete contin. Dyn. syst. 9, 651-662 (2003) · Zbl 1029.35180 · doi:10.3934/dcds.2003.9.651
[14]Sogge, C. D.: Lectures on nonlinear wave equations, Monogr. anal. 2 (1995) · Zbl 1089.35500
[15]Sugitani, Y.; Kawashima, S.: Decay estimates of solution to a semi-linear dissipative plate equation, J. hyperbolic differ. Equ. 7, 471-501 (2010) · Zbl 1207.35229 · doi:10.1142/S0219891610002207
[16]Todorova, G.; Yordanov, B.: Critical exponent for a nonlinear wave equation with damping, J. differential equations 174, 464-489 (2001) · Zbl 0994.35028 · doi:10.1006/jdeq.2000.3933
[17]Wang, S. -B.; Chen, G. -W.: Small amplitude solutions of the generalized imbq equation, J. math. Anal. appl. 274, 846-866 (2002) · Zbl 1136.35425 · doi:10.1016/S0022-247X(02)00401-8
[18]Wang, Y. -Z.: Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation, Nonlinear anal. 70, 465-482 (2009) · Zbl 1161.35470 · doi:10.1016/j.na.2007.12.018
[19]Wang, Y. -Z.: Global existence of classical solutions to the minimal surface equation in two space dimensions with slow decay initial value, J. math. Phys. 50, 103506 (2009)
[20]Wang, Y. -Z.; Wang, Y. -X.: Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order, Nonlinear anal. 72, 4500-4507 (2010) · Zbl 1189.35172 · doi:10.1016/j.na.2010.02.025
[21]Wang, Y. Z.; Wang, Y. X.: Global existence of classical solutions to the minimal surface equation with slow decay initial value, Appl. comput. Math. 216, 576-583 (2010) · Zbl 1184.49042 · doi:10.1016/j.amc.2010.01.079
[22]Wang, W. -K.; Wang, W. -J.: The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions, J. math. Anal. appl. 368, 226-241 (2010) · Zbl 1184.35218 · doi:10.1016/j.jmaa.2009.12.013
[23]Yang, Z. -J.: Longtime behavior of the Kirchhoff type equation with strong damping on, J. differential equations 242, 269-286 (2007) · Zbl 1208.35147 · doi:10.1016/j.jde.2007.08.004