Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation

Ghisi, Marina; Gobbino, Massimo; Haraux, Alain

doi:10.1090/tran/6520

Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation
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by Marina Ghisi, Massimo Gobbino and Alain Haraux PDF

Trans. Amer. Math. Soc. 368 (2016), 2039-2079 Request permission

Abstract:

We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the “elastic” operator.

In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup and the further regularity of solutions. In the non-homogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution.

What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term.

We also provide counterexamples in order to show the optimality of our results.

References

Luigi Amerio and Giovanni Prouse, Uniqueness and almost-periodicity theorems for a non linear wave equation, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 46 (1969), 1–8 (English, with Italian summary). MR 255993
Marco Biroli, Bounded or almost periodic solution of the non linear vibrating membrane equation, Ricerche Mat. 22 (1973), 190–202. MR 361399
Marco Biroli and Alain Haraux, Asymptotic behavior for an almost periodic, strongly dissipative wave equation, J. Differential Equations 38 (1980), no. 3, 422–440. MR 605059, DOI 10.1016/0022-0396(80)90017-0
Thierry Cazenave and Alain Haraux, An introduction to semilinear evolution equations, Oxford Lecture Series in Mathematics and its Applications, vol. 13, The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1990 French original by Yvan Martel and revised by the authors. MR 1691574
G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39 (1981/82), no. 4, 433–454. MR 644099, DOI 10.1090/S0033-569X-1982-0644099-3
Shu Ping Chen and Roberto Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math. 136 (1989), no. 1, 15–55. MR 971932
Shu Ping Chen and Roberto Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Differential Equations 88 (1990), no. 2, 279–293. MR 1081250, DOI 10.1016/0022-0396(90)90100-4
Shu Ping Chen and Roberto Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0<\alpha <\frac 12$, Proc. Amer. Math. Soc. 110 (1990), no. 2, 401–415. MR 1021208, DOI 10.1090/S0002-9939-1990-1021208-4
Ralph Chill and Alain Haraux, An optimal estimate for the difference of solutions of two abstract evolution equations, J. Differential Equations 193 (2003), no. 2, 385–395. MR 1998640, DOI 10.1016/S0022-0396(03)00057-3
M. D’Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci. 37 (2014), no. 11, 1570–1592. MR 3225191, DOI 10.1002/mma.2913
Luci Harue Fatori, Maria Zegarra Garay, and Jaime E. Muñoz Rivera, Differentiability, analyticity and optimal rates of decay for damped wave equations, Electron. J. Differential Equations (2012), No. 48, 13. MR 2927784
A. Haraux, Uniform decay and Lagrange stability for linear contraction semi-groups, Mat. Apl. Comput. 7 (1988), no. 3, 143–154 (English, with Portuguese summary). MR 994760
Alain Haraux and Mitsuharu Ôtani, Analyticity and regularity for a class of second order evolution equations, Evol. Equ. Control Theory 2 (2013), no. 1, 101–117. MR 3085628, DOI 10.3934/eect.2013.2.101
Alain Haraux, Nonlinear evolution equations—global behavior of solutions, Lecture Notes in Mathematics, vol. 841, Springer-Verlag, Berlin-New York, 1981. MR 610796
Alain Haraux, Damping out of transient states for some semilinear, quasiautonomous systems of hyperbolic type, Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 7 (1983), 89–136. MR 738758
Alain Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep. 3 (1987), no. 1, i–xxiv and 1–281. MR 1078761
Alain Haraux, Anti-periodic solutions of some nonlinear evolution equations, Manuscripta Math. 63 (1989), no. 4, 479–505. MR 991267, DOI 10.1007/BF01171760
Alain Haraux, Nonresonance for a strongly dissipative wave equation in higher dimensions, Manuscripta Math. 53 (1985), no. 1-2, 145–166. MR 804342, DOI 10.1007/BF01174015
A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal. 100 (1988), no. 2, 191–206. MR 913963, DOI 10.1007/BF00282203
Ryo Ikehata, Decay estimates of solutions for the wave equations with strong damping terms in unbounded domains, Math. Methods Appl. Sci. 24 (2001), no. 9, 659–670. MR 1834920, DOI 10.1002/mma.235
Ryo Ikehata and Masato Natsume, Energy decay estimates for wave equations with a fractional damping, Differential Integral Equations 25 (2012), no. 9-10, 939–956. MR 2985688
Ryo Ikehata, Grozdena Todorova, and Borislav Yordanov, Wave equations with strong damping in Hilbert spaces, J. Differential Equations 254 (2013), no. 8, 3352–3368. MR 3020879, DOI 10.1016/j.jde.2013.01.023
B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. MR 690064
Kangsheng Liu and Zhuangyi Liu, Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces, J. Differential Equations 141 (1997), no. 2, 340–355. MR 1488357, DOI 10.1006/jdeq.1997.3331
S. Matthes and M. Reissig, Qualitative properties of structural damped wave models, Eurasian Math. J. To appear.
Delio Mugnolo, A variational approach to strongly damped wave equations, Functional analysis and evolution equations, Birkhäuser, Basel, 2008, pp. 503–514. MR 2402747, DOI 10.1007/978-3-7643-7794-6_{3}0
Kenji Nishihara, Degenerate quasilinear hyperbolic equation with strong damping, Funkcial. Ekvac. 27 (1984), no. 1, 125–145. MR 763940
Kosuke Ono and Kenji Nishihara, On a nonlinear degenerate integro-differential equation of hyperbolic type with a strong dissipation, Adv. Math. Sci. Appl. 5 (1995), no. 2, 457–476. MR 1361000
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
Yoshihiro Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci. 23 (2000), no. 3, 203–226. MR 1736935, DOI 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M