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

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 coupledsystem. 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.