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Keywords:
nonexistence; test functions; global weak solution; fractional Laplacian; critical exponent
nonexistence; test functions; global weak solution; fractional Laplacian; critical exponent
Summary:
In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), $$ u_{tt} +f(x)u_t +(-\Delta)^{\alpha/2}(u^m)= h(t,x) |u|^{p}, $$ posed in $(0,T)\times \mathbb{R}^{N},$ where $(-\Delta)^{{\alpha}/{2}}, 0<\alpha \leq 2$ is ${\alpha}/{2}$-fractional power of $\,-\Delta.$ Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a $2\times2$ system of the same type.
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