American Institute of Mathematical Sciences

Advanced
December  2020, 13(12): 3525-3533. doi: 10.3934/dcdss.2020247

Large data solutions for semilinear higher order equations

Sandra Lucente

Dipartimento interateneo di Fisica, Università degli Studi di Bari, Via Orabona 4 70125 Bari, Italy

Received  March 2019 Revised  June 2019 Published  December 2020 Early access  January 2020

In this paper we study local and global in time existence for a class of nonlinear evolution equations having order eventually greater than 2 and not integer. The linear operator has an homogeneous damping term; the nonlinearity is of polynomial type without derivatives:
$ u_{tt}+ (-\Delta)^{2\theta}u+2\mu(-\Delta)^\theta u_t + |u|^{p-1}u = 0, \quad t\geq0, \ x\in {\mathbb{R}}^n, $
with
$ \mu>0 $
,
$ \theta>0 $
. Since we are treating an absorbing nonlinear term, large data solutions can be considered.
Keywords: Large data, damping wave, energy estimate.
Mathematics Subject Classification: Primary: 35L71, 35L82, 35B40.
Citation: Sandra Lucente. Large data solutions for semilinear higher order equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3525-3533. doi: 10.3934/dcdss.2020247
References:
[1]

M. D'Abbicco and M. R. Ebert, A classification of structural dissipations for evolution operators, Math. Methods Appl. Sci., 39 (2016), 2558-2582.  doi: 10.1002/mma.3713.  Google Scholar

[2]

M. D'Abbicco and M. R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations, Nonlinear Analysis, 149 (2017), 1-40.  doi: 10.1016/j.na.2016.10.010.  Google Scholar

[3]

M. D'Abbicco and S. Lucente, The beam equation with nonlinear memory, Zeitschrift fur Angewadte Mathematik und Physik, 67 (2016), Art. 60, 18 pp. doi: 10.1007/s00033-016-0655-x.  Google Scholar

[4]

M. D'Abbicco, M. R. Ebert and S. Lucente, Self-similar asymptotic profile of the solution to a nonlinear evolution equation with critical dissipation, Math Meth Appl Sci., 40 (2017), 6480-6494.  doi: 10.1002/mma.4469.  Google Scholar

[5]

H. Hajaiej, X. Yu and Z. Zhai, Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms, Journal of Mathematical Analysis and Applications, 396 (2012), 569-577.  doi: 10.1016/j.jmaa.2012.06.054.  Google Scholar

[6]

S. Lucente, Critical exponents and where to find them, Bruno Pini Mathematical Analysis Seminar, 9 (2018), 102-114.   Google Scholar

[7]

T. D. Pham, M. Kinane and M. Reissig, Global existence for semi-linear structurally damped $\sigma$-evolution models, J. Math. Anal. Appl., 431 (2015), 569-596.  doi: 10.1016/j.jmaa.2015.06.001.  Google Scholar

[8]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.  Google Scholar

show all references

References:
[1]

M. D'Abbicco and M. R. Ebert, A classification of structural dissipations for evolution operators, Math. Methods Appl. Sci., 39 (2016), 2558-2582.  doi: 10.1002/mma.3713.  Google Scholar

[2]

M. D'Abbicco and M. R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations, Nonlinear Analysis, 149 (2017), 1-40.  doi: 10.1016/j.na.2016.10.010.  Google Scholar

[3]

M. D'Abbicco and S. Lucente, The beam equation with nonlinear memory, Zeitschrift fur Angewadte Mathematik und Physik, 67 (2016), Art. 60, 18 pp. doi: 10.1007/s00033-016-0655-x.  Google Scholar

[4]

M. D'Abbicco, M. R. Ebert and S. Lucente, Self-similar asymptotic profile of the solution to a nonlinear evolution equation with critical dissipation, Math Meth Appl Sci., 40 (2017), 6480-6494.  doi: 10.1002/mma.4469.  Google Scholar

[5]

H. Hajaiej, X. Yu and Z. Zhai, Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms, Journal of Mathematical Analysis and Applications, 396 (2012), 569-577.  doi: 10.1016/j.jmaa.2012.06.054.  Google Scholar

[6]

S. Lucente, Critical exponents and where to find them, Bruno Pini Mathematical Analysis Seminar, 9 (2018), 102-114.   Google Scholar

[7]

T. D. Pham, M. Kinane and M. Reissig, Global existence for semi-linear structurally damped $\sigma$-evolution models, J. Math. Anal. Appl., 431 (2015), 569-596.  doi: 10.1016/j.jmaa.2015.06.001.  Google Scholar

[8]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.  Google Scholar

[1]

Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92
[2]

Ryo Ikehata, Shingo Kitazaki. Optimal energy decay rates for some wave equations with double damping terms. Evolution Equations & Control Theory, 2019, 8 (4) : 825-846. doi: 10.3934/eect.2019040
[3]

Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37
[4]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[5]

Marcello D'Abbicco. Small data solutions for semilinear wave equations with effective damping. Conference Publications, 2013, 2013 (special) : 183-191. doi: 10.3934/proc.2013.2013.183
[6]

Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133
[7]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064
[8]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45
[9]

Masataka Kato, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. Effect of energy-saving server scheduling on power consumption for large-scale data centers. Journal of Industrial & Management Optimization, 2016, 12 (2) : 667-685. doi: 10.3934/jimo.2016.12.667
[10]

Neal Bez, Chris Jeavons. A sharp Sobolev-Strichartz estimate for the wave equation. Electronic Research Announcements, 2015, 22: 46-54. doi: 10.3934/era.2015.22.46
[11]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883
[12]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
[13]

Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469
[14]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
[15]

Gisèle Ruiz Goldstein, Jerome A. Goldstein, Fabiana Travessini De Cezaro. Equipartition of energy for nonautonomous wave equations. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 75-85. doi: 10.3934/dcdss.2017004
[16]

Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407
[17]

Vando Narciso. On a Kirchhoff wave model with nonlocal nonlinear damping. Evolution Equations & Control Theory, 2020, 9 (2) : 487-508. doi: 10.3934/eect.2020021
[18]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165
[19]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015
[20]

Michael Herty, Lorenzo Pareschi, Giuseppe Visconti. Mean field models for large data–clustering problems. Networks & Heterogeneous Media, 2020, 15 (3) : 463-487. doi: 10.3934/nhm.2020027

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (156)
  • HTML views (330)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy