American Institute of Mathematical Sciences

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

Large data solutions for semilinear higher order equations

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

Received  March 2019 Revised  June 2019 Published  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.
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
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