# American Institute of Mathematical Sciences

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. Google Scholar [2] R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,", Second edition, 140 (2003). Google Scholar [3] V. Barbu, "Analysis and Control of Nonlinear Infinite Dimensional Systems,", Mathematics in Science and Engineering, 190 (1993). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical nonlinear wave equation,, J. Funct. Anal., 110 (1992), 96. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [20] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential and Integral Equations, 6 (1993), 507. Google Scholar [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). Google Scholar [22] V. Lakshmikantham and S. Leela, "Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations,", Mathematics in Science and Engineering, (1969). Google Scholar [23] G. Lebeau, Perte de régularité pour les équations d'ondes sur-critiques,, Bull. Soc. Math. France, 133 (2005), 145. Google Scholar [24] P.-K. Lin, "Köthe-Bochner Function Spaces,", Birkhäuser Boston, (2004). Google Scholar [25] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1,", Dunod, (1968). Google Scholar [26] L. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273. Google Scholar [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. Google Scholar [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. Google Scholar [29] P. Radu, Strong solutions for semilinear wave equations with damping and source terms,, Appl. Anal. Analysis, 92 (2013), 718. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### References:
 [1] J.-P. Aubin, Un théorème de compacité,, C. R. Acad. Sci. Paris, 256 (1963), 5042. Google Scholar [2] R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,", Second edition, 140 (2003). Google Scholar [3] V. Barbu, "Analysis and Control of Nonlinear Infinite Dimensional Systems,", Mathematics in Science and Engineering, 190 (1993). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical nonlinear wave equation,, J. Funct. Anal., 110 (1992), 96. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [20] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential and Integral Equations, 6 (1993), 507. Google Scholar [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). Google Scholar [22] V. Lakshmikantham and S. Leela, "Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations,", Mathematics in Science and Engineering, (1969). Google Scholar [23] G. Lebeau, Perte de régularité pour les équations d'ondes sur-critiques,, Bull. Soc. Math. France, 133 (2005), 145. Google Scholar [24] P.-K. Lin, "Köthe-Bochner Function Spaces,", Birkhäuser Boston, (2004). Google Scholar [25] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1,", Dunod, (1968). Google Scholar [26] L. Payne and D. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273. Google Scholar [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. Google Scholar [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. Google Scholar [29] P. Radu, Strong solutions for semilinear wave equations with damping and source terms,, Appl. Anal. Analysis, 92 (2013), 718. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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