June  2013, 2(2): 255-279. doi: 10.3934/eect.2013.2.255

Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping

1. 

NC State University, Department of Mathematics, 3236 SAS Hall, Raleigh, NC 27695-8205

2. 

Department of Mathematics, University of Nebraska-Lincoln, Avery Hall 239, Lincoln, NE 68588

3. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588

Received  January 2013 Revised  February 2013 Published  March 2013

We study regular solutions to wave equations with super-critical source terms, e.g., of exponent $p>5$ in 3D. Such sources have been a major challenge in the investigation of finite-energy ($H^1 \times L^2$) solutions to wave PDEs for many years. The wellposedness has been settled in part, but even the local existence, for instance, in 3 dimensions requires the relation $p\leq 6m/(m+1)$ between the exponents $p$ of the source and $m$ of the viscous damping.
    We prove that smooth initial data ($H^2 \times H^1$) yields regular solutions that do not depend on the above correlation. Local existence is demonstrated for any source exponent $p\geq 1$ and any monotone damping including feedbacks growing exponentially or logarithmically at infinity, or with no damping at all. The result holds in dimensions 3 and 4, and with some restrictions on $p$ in dimensions $n\geq 5$. Furthermore, if we assert the classical condition that the damping grows as fast as the source, then these regular solutions are global.
Citation: Lorena Bociu, Petronela Radu, Daniel Toundykov. Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping. Evolution Equations and Control Theory, 2013, 2 (2) : 255-279. doi: 10.3934/eect.2013.2.255
References:
[1]

J.-P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044.

[2]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.

[3]

V. Barbu, "Analysis and Control of Nonlinear Infinite Dimensional Systems," Mathematics in Science and Engineering, 190, Academic Press, Inc., Boston, MA, 1993.

[4]

V. Barbu, I. Lasiecka and M. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Trans. Amer. Math. Soc., 357 (2005), 2571-2611 (electronic). doi: 10.1090/S0002-9947-05-03880-8.

[5]

L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping, Nonlinear Analysis A: Theory, Methods and Applications, 71 (2009), e560-e575. doi: 10.1016/j.na.2008.11.062.

[6]

L. Bociu and I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Applicationes Mathematicae, 35 (2008), 281-304. doi: 10.4064/am35-3-3.

[7]

L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860. doi: 10.3934/dcds.2008.22.835.

[8]

L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683. doi: 10.1016/j.jde.2010.03.009.

[9]

L. Bociu and P. Radu, Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping, Discrete Contin. Dyn. Syst., 2009, Dynamical Systems, Differential Equations and Applications, 7th AIMS Conference, suppl., 60-71.

[10]

L. Bociu, M. Rammaha and D. Toundykov, On a wave equation with supercritical interior and boundary sources and damping terms, Math. Nachr., 284 (2011), 2032-2064. doi: 10.1002/mana.200910182.

[11]

M. Cavalcanti, V. N. Cavalcanti and P. Martinez, Existence and decay rates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158. doi: 10.1016/j.jde.2004.04.011.

[12]

I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Communications in Partial Differential Equations, 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132.

[13]

I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics," Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[14]

E. Fereisl, Global attractors for semilinear damped wave equations with supercritical exponent, Journal of Differential Equations, 116 (1995), 431-447. doi: 10.1006/jdeq.1995.1042.

[15]

J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal., 110 (1992), 96-130.

[16]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, Journal of Differential Equations, 109 (1994), 295-308. doi: 10.1006/jdeq.1994.1051.

[17]

W. Gong and Z. Shi, Drop properties and approximative compactness in Orlicz-Bochner function spaces, J. Math. Anal. Appl., 344 (2008), 748-756. doi: 10.1016/j.jmaa.2008.03.024.

[18]

A. Kamińska, Some convexity properties of Musielak-Orlicz spaces of Bochner type, Proceedings of the 13th Winter School on Abstract Analysis (Srní, 1985), Rend. Circ. Mat. Palermo (2) Suppl., 10 (1985), 63-73 (1986).

[19]

M. A. Krasnosel'skiĭ and J. B. Rutickiĭ, "Convex Functions and Orlicz Spaces," Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961.

[20]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential and Integral Equations, 6 (1993), 507-533.

[21]

I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems," Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000.

[22]

V. Lakshmikantham and S. Leela, "Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations," Mathematics in Science and Engineering, Vol. 55-I, Academic Press, New York, 1969.

[23]

G. Lebeau, Perte de régularité pour les équations d'ondes sur-critiques, Bull. Soc. Math. France, 133 (2005), 145-157.

[24]

P.-K. Lin, "Köthe-Bochner Function Spaces," Birkhäuser Boston, Inc., Boston, MA, 2004.

[25]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1," Dunod, Paris, 1968.

[26]

L. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.

[27]

P. Radu, Weak solutions to the Cauchy problem of a semilinear wave equation with damping and source terms, Advances in Differential Equations, 10 (2005), 1261-1300.

[28]

P. Radu, Weak solutions to the initial boundary value problem of a semilinear wave equation with damping and source terms, Applicationae Mathematica (Warsaw), 35 (2008), 355-378. doi: 10.4064/am35-3-7.

[29]

P. Radu, Strong solutions for semilinear wave equations with damping and source terms, Appl. Anal. Analysis, 92 (2013), 718-739.

[30]

J. Serrin, G. Todorova and E. Vitillaro, Existence for a nonlinear wave equation with damping and source terms, Differential Integral Equations, 16 (2003), 13-50.

[31]

S. Shang, Y. Cui and Y. Fu, Nearly strict convexity in Musielak-Orlicz-Bochner function spaces, Nonlinear Anal., 74 (2011), 6333-6341. doi: 10.1016/j.na.2011.06.013.

[32]

J. Simon, Compact sets in the space $L_p(0,T;B)$, Annali di Mat. Pura et Applicate (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[33]

G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, Nonlinear Anal., 41 (2000), 891-905. doi: 10.1016/S0362-546X(98)00317-4.

[34]

G. Todorova and E. Vitillaro, Blow-up for nonlinear dissipative wave equations in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 303 (2005), 242-257. doi: 10.1016/j.jmaa.2004.08.039.

[35]

E. Vitillaro, Global existence of the wave equation with nonlinear boundary damping and source terms, J. of Differential Equations, 186 (2002), 259-298. doi: 10.1016/S0022-0396(02)00023-2.

show all references

References:
[1]

J.-P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044.

[2]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.

[3]

V. Barbu, "Analysis and Control of Nonlinear Infinite Dimensional Systems," Mathematics in Science and Engineering, 190, Academic Press, Inc., Boston, MA, 1993.

[4]

V. Barbu, I. Lasiecka and M. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Trans. Amer. Math. Soc., 357 (2005), 2571-2611 (electronic). doi: 10.1090/S0002-9947-05-03880-8.

[5]

L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping, Nonlinear Analysis A: Theory, Methods and Applications, 71 (2009), e560-e575. doi: 10.1016/j.na.2008.11.062.

[6]

L. Bociu and I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Applicationes Mathematicae, 35 (2008), 281-304. doi: 10.4064/am35-3-3.

[7]

L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860. doi: 10.3934/dcds.2008.22.835.

[8]

L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683. doi: 10.1016/j.jde.2010.03.009.

[9]

L. Bociu and P. Radu, Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping, Discrete Contin. Dyn. Syst., 2009, Dynamical Systems, Differential Equations and Applications, 7th AIMS Conference, suppl., 60-71.

[10]

L. Bociu, M. Rammaha and D. Toundykov, On a wave equation with supercritical interior and boundary sources and damping terms, Math. Nachr., 284 (2011), 2032-2064. doi: 10.1002/mana.200910182.

[11]

M. Cavalcanti, V. N. Cavalcanti and P. Martinez, Existence and decay rates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158. doi: 10.1016/j.jde.2004.04.011.

[12]

I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Communications in Partial Differential Equations, 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132.

[13]

I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics," Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[14]

E. Fereisl, Global attractors for semilinear damped wave equations with supercritical exponent, Journal of Differential Equations, 116 (1995), 431-447. doi: 10.1006/jdeq.1995.1042.

[15]

J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal., 110 (1992), 96-130.

[16]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, Journal of Differential Equations, 109 (1994), 295-308. doi: 10.1006/jdeq.1994.1051.

[17]

W. Gong and Z. Shi, Drop properties and approximative compactness in Orlicz-Bochner function spaces, J. Math. Anal. Appl., 344 (2008), 748-756. doi: 10.1016/j.jmaa.2008.03.024.

[18]

A. Kamińska, Some convexity properties of Musielak-Orlicz spaces of Bochner type, Proceedings of the 13th Winter School on Abstract Analysis (Srní, 1985), Rend. Circ. Mat. Palermo (2) Suppl., 10 (1985), 63-73 (1986).

[19]

M. A. Krasnosel'skiĭ and J. B. Rutickiĭ, "Convex Functions and Orlicz Spaces," Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961.

[20]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential and Integral Equations, 6 (1993), 507-533.

[21]

I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems," Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000.

[22]

V. Lakshmikantham and S. Leela, "Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations," Mathematics in Science and Engineering, Vol. 55-I, Academic Press, New York, 1969.

[23]

G. Lebeau, Perte de régularité pour les équations d'ondes sur-critiques, Bull. Soc. Math. France, 133 (2005), 145-157.

[24]

P.-K. Lin, "Köthe-Bochner Function Spaces," Birkhäuser Boston, Inc., Boston, MA, 2004.

[25]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1," Dunod, Paris, 1968.

[26]

L. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.

[27]

P. Radu, Weak solutions to the Cauchy problem of a semilinear wave equation with damping and source terms, Advances in Differential Equations, 10 (2005), 1261-1300.

[28]

P. Radu, Weak solutions to the initial boundary value problem of a semilinear wave equation with damping and source terms, Applicationae Mathematica (Warsaw), 35 (2008), 355-378. doi: 10.4064/am35-3-7.

[29]

P. Radu, Strong solutions for semilinear wave equations with damping and source terms, Appl. Anal. Analysis, 92 (2013), 718-739.

[30]

J. Serrin, G. Todorova and E. Vitillaro, Existence for a nonlinear wave equation with damping and source terms, Differential Integral Equations, 16 (2003), 13-50.

[31]

S. Shang, Y. Cui and Y. Fu, Nearly strict convexity in Musielak-Orlicz-Bochner function spaces, Nonlinear Anal., 74 (2011), 6333-6341. doi: 10.1016/j.na.2011.06.013.

[32]

J. Simon, Compact sets in the space $L_p(0,T;B)$, Annali di Mat. Pura et Applicate (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[33]

G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, Nonlinear Anal., 41 (2000), 891-905. doi: 10.1016/S0362-546X(98)00317-4.

[34]

G. Todorova and E. Vitillaro, Blow-up for nonlinear dissipative wave equations in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 303 (2005), 242-257. doi: 10.1016/j.jmaa.2004.08.039.

[35]

E. Vitillaro, Global existence of the wave equation with nonlinear boundary damping and source terms, J. of Differential Equations, 186 (2002), 259-298. doi: 10.1016/S0022-0396(02)00023-2.

[1]

Lorena Bociu, Petronela Radu, Daniel Toundykov. Errata: Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping. Evolution Equations and Control Theory, 2014, 3 (2) : 349-354. doi: 10.3934/eect.2014.3.349

[2]

Tomasz Dlotko, Tongtong Liang, Yejuan Wang. Critical and super-critical abstract parabolic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1517-1541. doi: 10.3934/dcdsb.2019238

[3]

Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459

[4]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307

[5]

A. M. Micheletti, Monica Musso, A. Pistoia. Super-position of spikes for a slightly super-critical elliptic equation in $R^N$. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 747-760. doi: 10.3934/dcds.2005.12.747

[6]

Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure and Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921

[7]

A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119

[8]

Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351

[9]

Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090

[10]

Patrick Martinez, Jean-Michel Roquejoffre. The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2445-2472. doi: 10.3934/cpaa.2012.11.2445

[11]

Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377

[12]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[13]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[14]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179

[15]

Ting Guo, Xianhua Tang, Qi Zhang, Zu Gao. Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1563-1579. doi: 10.3934/cpaa.2020078

[16]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795

[17]

Juncheng Wei, Ke Wu. Local behavior of solutions to a fractional equation with isolated singularity and critical Serrin exponent. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4031-4050. doi: 10.3934/dcds.2022044

[18]

Chunyan Zhao, Chengkui Zhong, Zhijun Tang. Asymptotic behavior of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022025

[19]

Björn Birnir, Kenneth Nelson. The existence of smooth attractors of damped and driven nonlinear wave equations with critical exponent , s = 5. Conference Publications, 1998, 1998 (Special) : 100-117. doi: 10.3934/proc.1998.1998.100

[20]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231

2021 Impact Factor: 1.169

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]