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 & 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.

[2]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,", Second edition, 140 (2003).

[3]

V. Barbu, "Analysis and Control of Nonlinear Infinite Dimensional Systems,", Mathematics in Science and Engineering, 190 (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. 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, 71 (2009). 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. 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. 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. 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 (): 60.

[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. 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. 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. 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, (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. 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.

[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. 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. 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, 10 (1985), 63.

[19]

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

[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 (2000).

[22]

V. Lakshmikantham and S. Leela, "Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations,", Mathematics in Science and Engineering, (1969).

[23]

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

[24]

P.-K. Lin, "Köthe-Bochner Function Spaces,", Birkhäuser Boston, (2004).

[25]

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

[26]

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

[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.

[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. 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.

[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.

[31]

S. Shang, Y. Cui and Y. Fu, Nearly strict convexity in Musielak-Orlicz-Bochner function spaces,, Nonlinear Anal., 74 (2011), 6333. 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. 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. doi: 10.1016/S0362-546X(98)00317-4.

[34]

G. Todorova and E. Vitillaro, Blow-up for nonlinear dissipative wave equations in $\mathbbR^n$,, J. Math. Anal. Appl., 303 (2005), 242. 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. 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.

[2]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,", Second edition, 140 (2003).

[3]

V. Barbu, "Analysis and Control of Nonlinear Infinite Dimensional Systems,", Mathematics in Science and Engineering, 190 (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. 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, 71 (2009). 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. 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. 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. 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 (): 60.

[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. 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. 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. 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, (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. 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.

[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. 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. 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, 10 (1985), 63.

[19]

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

[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 (2000).

[22]

V. Lakshmikantham and S. Leela, "Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations,", Mathematics in Science and Engineering, (1969).

[23]

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

[24]

P.-K. Lin, "Köthe-Bochner Function Spaces,", Birkhäuser Boston, (2004).

[25]

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

[26]

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

[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.

[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. 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.

[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.

[31]

S. Shang, Y. Cui and Y. Fu, Nearly strict convexity in Musielak-Orlicz-Bochner function spaces,, Nonlinear Anal., 74 (2011), 6333. 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. 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. doi: 10.1016/S0362-546X(98)00317-4.

[34]

G. Todorova and E. Vitillaro, Blow-up for nonlinear dissipative wave equations in $\mathbbR^n$,, J. Math. Anal. Appl., 303 (2005), 242. 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. 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 & Control Theory, 2014, 3 (2) : 349-354. doi: 10.3934/eect.2014.3.349

[2]

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 & Continuous Dynamical Systems - A, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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 & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090

[9]

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

[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 & Applied Analysis, 2012, 11 (6) : 2445-2472. doi: 10.3934/cpaa.2012.11.2445

[11]

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

[12]

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

[13]

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

[14]

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

[15]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022

[16]

Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217

[17]

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

[18]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251

[19]

Michinori Ishiwata. Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent. Conference Publications, 2005, 2005 (Special) : 443-452. doi: 10.3934/proc.2005.2005.443

[20]

Y. Kabeya. Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 117-134. doi: 10.3934/dcds.2006.14.117

2017 Impact Factor: 1.049

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]