Remarks on the linear wave equation
Author:
John M. Ball
Journal:
Quart. Appl. Math. 82 (2024), 431-448
MSC (2020):
Primary 35L05, 47D06; Secondary 42B37
DOI:
https://doi.org/10.1090/qam/1678
Published electronically:
October 5, 2023
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We make some remarks on the linear wave equation concerning the existence and uniqueness of weak solutions, satisfaction of the energy equation, growth properties of solutions, the passage from bounded to unbounded domains, and reconciliation of different representations of solutions.
References
- Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2nd ed., Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2798103, DOI 10.1007/978-3-0348-0087-7
- A. V. Balakrishnan, Applied functional analysis, 2nd ed., Applications of Mathematics, vol. 3, Springer-Verlag, New York-Berlin, 1981. MR 612793
- J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977), no. 2, 370–373. MR 442748, DOI 10.1090/S0002-9939-1977-0442748-6
- J. M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations.ations, J. Differential Equations 27 (1978), no. 2, 224–265. MR 461576, DOI 10.1016/0022-0396(78)90032-3
- J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 31–52. Partial differential equations and applications. MR 2026182, DOI 10.3934/dcds.2004.10.31
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829, DOI 10.1007/978-0-387-70914-7
- A. R. Brodsky, On the asymptotic behavior of solutions of the wave equations, Proc. Amer. Math. Soc. 18 (1967), 207–208. MR 212417, DOI 10.1090/S0002-9939-1967-0212417-X
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- H. O. Fattorini, Second order linear differential equations in Banach spaces, North-Holland Mathematics Studies, vol. 108, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 99. MR 797071
- Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 200692
- Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
- J. A. Goldstein and M. Wacker, The energy space and norm growth for abstract wave equations, Appl. Math. Lett. 16 (2003), no. 5, 767–772. MR 1986048, DOI 10.1016/S0893-9659(03)00080-6
- Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437, DOI 10.1007/978-0-387-09432-8
- Sigurdur Helgason, The Radon transform, 2nd ed., Progress in Mathematics, vol. 5, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1723736, DOI 10.1007/978-1-4757-1463-0
- C. S. Herz, Fourier transforms related to convex sets, Ann. of Math. (2) 75 (1962), 81–92. MR 142978, DOI 10.2307/1970421
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, RI, 1957. rev. ed. MR 89373
- Edmund Hlawka, Über Integrale auf konvexen Körpern. I, Monatsh. Math. 54 (1950), 1–36 (German). MR 37003, DOI 10.1007/BF01304101
- Taqdir Husain, The open mapping and closed graph theorems in topological vector spaces, Clarendon Press, Oxford, 1965. MR 178331, DOI 10.1007/978-3-322-96210-2
- Ryo Ikehata, $L^2$-blowup estimates of the wave equation and its application to local energy decay, J. Hyperbolic Differ. Equ. 20 (2023), no. 1, 259–275. MR 4591582, DOI 10.1142/S021989162350008X
- Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 75429
- Fritz John, Partial differential equations, 4th ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1982. MR 831655, DOI 10.1007/978-1-4684-9333-7
- Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 407617
- Peter D. Lax, Hyperbolic partial differential equations, Courant Lecture Notes in Mathematics, vol. 14, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2006. With an appendix by Cathleen S. Morawetz. MR 2273657, DOI 10.1090/cln/014
- Boris Makarov and Anatolii Podkorytov, Real analysis: measures, integrals and applications, Universitext, Springer, London, 2013. Translated from the 2011 Russian original. MR 3089088, DOI 10.1007/978-1-4471-5122-7
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- George R. Sell and Yuncheng You, Dynamics of evolutionary equations, Applied Mathematical Sciences, vol. 143, Springer-Verlag, New York, 2002. MR 1873467, DOI 10.1007/978-1-4757-5037-9
- Marvin Shinbrot, Asymptotic behavior of solutions of abstract wave equations, Proc. Amer. Math. Soc. 19 (1968), 1403–1406. MR 231246, DOI 10.1090/S0002-9939-1968-0231246-5
- Elias M. Stein, Maximal functions: Poisson integrals on symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 8, 2547–2549. MR 420118, DOI 10.1073/pnas.73.8.2547
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, NJ, 1971. MR 304972
- François Trèves, Basic linear partial differential equations, Pure and Applied Mathematics, Vol. 62, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 447753
- Junyan Zhao and Dashan Fan, Derivative estimates of averaging operators and extension, Front. Math. China 14 (2019), no. 2, 475–491. MR 3969117, DOI 10.1007/s11464-019-0755-y
References
- Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2nd ed., Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2798103, DOI 10.1007/978-3-0348-0087-7
- A. V. Balakrishnan, Applied functional analysis, 2nd ed., Applications of Mathematics, vol. 3, Springer-Verlag, New York-Berlin, 1981. MR 612793
- J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977), no. 2, 370–373. MR 442748, DOI 10.2307/2041821
- J. M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations.ations, J. Differential Equations 27 (1978), no. 2, 224–265. MR 461576, DOI 10.1016/0022-0396(78)90032-3
- J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 31–52. MR 2026182, DOI 10.3934/dcds.2004.10.31
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
- A. R. Brodsky, On the asymptotic behavior of solutions of the wave equations, Proc. Amer. Math. Soc. 18 (1967), 207–208. MR 212417, DOI 10.2307/2035261
- Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
- H. O. Fattorini, Second order linear differential equations in Banach spaces, North-Holland Mathematics Studies, vol. 108, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 99. MR 797071
- Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 200692
- Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
- J. A. Goldstein and M. Wacker, The energy space and norm growth for abstract wave equations, Appl. Math. Lett. 16 (2003), no. 5, 767–772. MR 1986048, DOI 10.1016/S0893-9659(03)00080-6
- Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
- Sigurdur Helgason, The Radon transform, 2nd ed., Progress in Mathematics, vol. 5, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1723736, DOI 10.1007/978-1-4757-1463-0
- C. S. Herz, Fourier transforms related to convex sets, Ann. of Math. (2) 75 (1962), 81–92. MR 142978, DOI 10.2307/1970421
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, RI, 1957. rev. ed. MR 89373
- Edmund Hlawka, Über Integrale auf konvexen Körpern. I, Monatsh. Math. 54 (1950), 1–36 (German). MR 37003, DOI 10.1007/BF01304101
- Taqdir Husain, The open mapping and closed graph theorems in topological vector spaces, Clarendon Press, Oxford, 1965. MR 178331
- Ryo Ikehata, $L^2$-blowup estimates of the wave equation and its application to local energy decay, J. Hyperbolic Differ. Equ. 20 (2023), no. 1, 259–275. MR 4591582, DOI 10.1142/S021989162350008X
- Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 75429
- Fritz John, Partial differential equations, 4th ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1982. MR 831655, DOI 10.1007/978-1-4684-9333-7
- Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 407617
- Peter D. Lax, Hyperbolic partial differential equations, Courant Lecture Notes in Mathematics, vol. 14, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2006. With an appendix by Cathleen S. Morawetz. MR 2273657, DOI 10.1090/cln/014
- Boris Makarov and Anatolii Podkorytov, Real analysis: measures, integrals and applications, Universitext, Springer, London, 2013. Translated from the 2011 Russian original. MR 3089088, DOI 10.1007/978-1-4471-5122-7
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- George R. Sell and Yuncheng You, Dynamics of evolutionary equations, Applied Mathematical Sciences, vol. 143, Springer-Verlag, New York, 2002. MR 1873467, DOI 10.1007/978-1-4757-5037-9
- Marvin Shinbrot, Asymptotic behavior of solutions of abstract wave equations, Proc. Amer. Math. Soc. 19 (1968), 1403–1406. MR 231246, DOI 10.2307/2036223
- Elias M. Stein, Maximal functions: Poisson integrals on symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 8, 2547–2549. MR 420118, DOI 10.1073/pnas.73.8.2547
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, NJ, 1971. MR 304972
- François Trèves, Basic linear partial differential equations, Pure and Applied Mathematics, Vol. 62, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 447753
- Junyan Zhao and Dashan Fan, Derivative estimates of averaging operators and extension, Front. Math. China 14 (2019), no. 2, 475–491. MR 3969117, DOI 10.1007/s11464-019-0755-y
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2020):
35L05,
47D06,
42B37
Retrieve articles in all journals
with MSC (2020):
35L05,
47D06,
42B37
Additional Information
John M. Ball
Affiliation:
Department of Mathematics, Heriot-Watt University, and Maxwell Institute for Mathematical Sciences, Edinburgh, United Kingdom; Hong Kong Institute for Advanced Study, City University of Hong Kong, Kowloon Tong, Hong Kong
MR Author ID:
30065
ORCID:
0000-0003-2496-2020
Email:
jb101@hw.ac.uk
Received by editor(s):
July 30, 2023
Published electronically:
October 5, 2023
Dedicated:
Dedicated to Bob Pego in admiration
Article copyright:
© Copyright 2023
Brown University