NONEXISTENCE RESULTS FOR SEMI-LINEAR MOORE-GIBSON-THOMPSON EQUATION WITH NON LOCAL OPERATOR

Yıl 2022, Cilt: 6 Sayı: 2, 191 - 201, 30.06.2022

Hakem Alı , Svetlin Georgiev

https://doi.org/10.31197/atnaa.947937

Öz

We study the nonexistence of global weak solutions to the following semi-linear Moore - Gibson-
Thompson equation with the nonlinearity of derivative type, namely,
$$
\left\{
\begin{array}{l}
u_{ttt}+u_{tt}-\Delta u-(-\Delta )^{\frac{\alpha}{2}}u_{t}
=|u_t|^p,\quad x\in \R^n,\quad t>0,\\
u(0,x)= u_0(x),\quad u_t(0,x)=u_1(x), \quad u_{tt}(0,x)= u_2(x) \quad x\in \R^n,
\end{array}
\right.
$$
where $\alpha\in (0, 2],\quad p> 1,$ and $(-\Delta)^{\frac{\alpha}{2}}$ is the fractional Laplacian operator of order $\frac{\alpha}{2}$. Then, this result is extended to the case of a weakly coupled
system. We intend to apply the method of a modified test function to establish nonexistence results and to overcome some difficulties as well caused by the well-known fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$.The results obtained in this paper extend several contributions in this field.

Anahtar Kelimeler

Test functions, nonexistence, lifespan estimates

Kaynakça

• [1] M. D'Abbicco, A note on a weakly coupled system of structurally damped waves, Dynamical Systems, Differential Equations and Applications, AIMS Proceedings. (2015) 320-329.
• [2] M. D'Abbicco, M. Reissig, Semi-linear structural damped waves, Math. Methods Appl. Sci. 37 (2014) 1570-1592.
• [3] M. D'Abbicco, M.R. Ebert, A classifiation of structural dissipations for evolution operators, Math. Methods Appl. Sci. 39 (2016) 2558-2582.
• [4] M. D'Abbicco, M.R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations, Nonlinear Anal. 149 (2017) 1-40.
• [5] L. Ca?arelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations. 32 (2007) 1245-1260.
• [6] W. Chen, R. Ikehata, The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case, J. Differential Equations. 292 (2021) 176-219.
• [7] W. Chen, A. Palmieri, A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case, Evol. Equ. Control Theory. (2020) 1-15. DOI: 10.3934/eect.2020085.
• [8] W. Chen, A. Palmieri, Nonexistence of global solutions for the semi-linear Moore-Gibson-Thompson equation in the conservative case, Discrete Contin. Dyn. Syst. 40(9) (2020) 5513-5540.
• [9] T.A. Dao, Global existence of solutions for weakly coupled systems of semi-linear structurally damped σ− evolution models with different power nonlinearities, 30A4, submitted.
• [10] T.A. Dao, Existence and nonexistence of global solutions for a structurally damped wave system with power nonlineari- ties,arXiv: 1911.04412v1, 2019.
• [11] T.A Dao, Ahmad Z. Fino, Critical exponent for semi-linear structurally damped wave equation of derivative type, arXiv: 2004.08486v2, 2020.
• [12] T.A. Dao, M. Reissig, A blow-up result for semi-linear structurally damped σ2 -evolution equations, preprint on arXiv:1909.01181v1, 2019.
• [13] T. Hadj Kaddour, A. Hakem Sufficient conditions of non global solution for fractional damped wave equations with non-linear memory, Advances in the Theory of Nonlinear Analysis and its Applications. 2(4) (2018) 224-237.
• [14] B. Kaltenbacher, I. Lasiecka, Exponential decay for low and higher energies in the third order linear Moore-Gibson- Thompson equation with variable viscosity, Palest. J. Math. 1 (1) (2012) 1-10.
• [15] M. Kwasnicki, Ten equivalent definitions of the fractional Laplace operator, Frac.Calc.Appl.Anal. 20(2017) 7-51.
• [16] M. Pellicer, B. Said-Houari, Wellposedness and decay rates for the Cauchy problem of the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Appl. Math. Optim. 80(2) (2019) 447-478.
• [17] R. Racke, B. Said-Houari, Global well-posedness of the Cauchy problem for the 3D Jordan- Moore-Gibson-Thompson equation, Commun. Contemp. Math., in press, (2021).
• [18] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (1) (2007) 67-112.