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Large data solutions for semilinear higher order equations

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  • 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.

    Mathematics Subject Classification: Primary: 35L71, 35L82, 35B40.

    Citation:

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  • [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.
    [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.
    [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.
    [4] M. D'AbbiccoM. 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.
    [5] H. HajaiejX. 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.
    [6] S. Lucente, Critical exponents and where to find them, Bruno Pini Mathematical Analysis Seminar, 9 (2018), 102-114. 
    [7] T. D. PhamM. 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.
    [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.
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