Abstract: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.
Keywords: nonexistence; test functions; global weak solution; fractional Laplacian; critical exponent
DOI: DOI 10.14712/1213-7243.2019.001
AMS Subject Classification: 47J35 35A01 35D30