Dynamics of a fluid equation with Neumann boundary conditions

We study the dynamics of a Neumann boundary value problem arising in fluid dynamics. We prove the nonexistence, existence and uniqueness of positive solutions under suitable conditions. At the same time, under stricter conditions, we also obtain the dynamic properties of the Neumann boundary value problem, such as the stability and instability of positive solutions. The methods of proof mainly involve the upper and lower solutions method, eigenvalue theory and some analysis techniques.

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