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In this paper, we consider the Boussinesq equations in a semi-infinite channel. We show that, under appropriate restrictions on the data, if the fluid velocity initially is small, the solution will tend exponentially to a transient laminar flow as the distance from the entry section tends to infinity. We also derive the explicit decay bounds.

In this paper, we study the blow-up phenomenon for a general nonlinear nonlocal porous medium equation in a bounded convex domain (Ω ∈ R n , n ≥ 3) with smooth boundary. Using the technique of a differential inequality and a Sobolev inequality, we derive the lower bound for the blow-up time under the nonlinear boundary condition if blow-up does really occur.

In this paper, we study the blow-up phenomenon for a nonlinear reaction-diffusion system with time-dependent coefficients under nonlinear boundary conditions. Using the technique of a first-order differential inequality and the Sobolev inequalities, we can get the energy expression which satisfies the differential inequality. The lower bound for the blow-up time could be obtained if blow-up does really occur in high dimensions.