Abstract
Sufficient conditions for stability, global asymptotic stability, and explosive instability are established for a class of nonlinear evolutional equations defined in Hilbert spaces by using certain relations between an abstract function and its Gateaux differential. These results are applied to specific forms of nonlinear evolutional equations arising from physics, in particular, a finite-dimensional system of complex ordinary differential equations, functional differential equations, and systems of complex partial differential equations describing nonlinear diffusion or wave phenomena.
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