- Advances in the Theory of Nonlinear Analysis and its Application
- Vol: 6 Issue: 2
- NONEXISTENCE RESULTS FOR SEMI-LINEAR MOORE-GIBSON-THOMPSON EQUATION WITH NON LOCAL OPERATOR
NONEXISTENCE RESULTS FOR SEMI-LINEAR MOORE-GIBSON-THOMPSON EQUATION WITH NON LOCAL OPERATOR
Authors : Hakem ALI, Svetlin GEORGİEV
Pages : 191-201
Doi:10.31197/atnaa.947937
View : 3 | Download : 1
Publication Date : 2022-06-30
Article Type : Research
Abstract :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.Keywords : Test functions, nonexistence, lifespan estimates