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.
📌 3 Key Takeaways
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Objective & Hypothesis: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 $\frac{\alpha}{2}$.
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Key Findings & Data:Then, this result is extended to the case of a weakly coupled system.
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Conclusion & Impact: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.
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.
Alı, H., Georgiev, S. (2022). NONEXISTENCE RESULTS FOR SEMI-LINEAR MOORE-GIBSON-THOMPSON EQUATION WITH NON LOCAL OPERATOR. Advances in the Theory of Nonlinear Analysis and its Application, Cilt: 6 Sayı: 2, 191–201. https://doi.org/10.31197/atnaa.947937
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