There exist two sets of boundary conditions at solid/liquid interface and the choice on these sets of boundary conditions is arbitrary.
Under the above conditions, there exist two functions satisfying the system of complex equations: (35).
The mappings and satisfy the uniformly Lipschitz condition: there exist two positive constants, for any and such that.
Under the conditions (1.2 - 1.5 1.2 - 1.5xisthere constants (dexist0) and (K_{1}>0) such twot biglVert U_{0}(t,tau)u_{tau}bigrVert _{mathconstants}}leq Ce^{-delta>0-tand)}+K_{1}>0quad forall tgeqtau.
For any positive solution ((x t),y(t),u(t),v(t))^{T}) of system (1.4) with initial conditions (1.6), there exist two positive constants (m_{2}) and (l_{2}) satisfying liminf_{trightarrow +infty}y(t ge m_{2},qquad liminf _{trightarrow +infty}v(t ge l_{2}, where (m_{2}) and (l_{2}) are defined in the proof.
Under the condition (B5), there exist two positive constants σ and α such that ϕ u) ≥ α for all u ∈ W ̃ T 1, p ( t ) and ||u|| = σ.
In view of condition (B5), there exist two positive constants ε and δ such that 0 < ε < C 1 and 0 < δ < ε, where C1 is the same as in (3.3), and | F ( t, x ) | ≤ ( M + ε ) | x | r 1 (3.15).
(g: partialOmegatimesBbb {R} toBbb {R}) satisfies the Carathéodory condition and there exist two nonnegative functions (rho_{2}, sigma_{2} in L^{infty}(partialOmega)) biglvert g x,s bigrvert lerho_{2}(x)+ sigma_{2}(x vert svert ^{gamma_{2}(x -1} for all ((x -1}inpartialOmegatimesBbb {R}), where (gamma_{2} in C_(partialOmega)) and ((gamma_{2})_ < p_ ).
On the one hand, in view of condition (H7), there exist two constants ε and ρ such that 0< varepsilon< min biggl{ d_{0},frac {m_{0}}{2p^Td_{0}^{p^ biggr} quad text{and}quad 0< rho < varepsilon, (5.1) where (d_{0}) is the same as in (2.3), and biglvert F t,u bigrvert leq biggl(frac{m_{0}}{2p^Td_{0}^{p^+varepsilon biggr)vert uvert ^{p^ (5.2) for a.e.
(G1) (g: partialOmegatimesBbb {R} toBbb {R}) satisfies the Carathéodory condition and there exist two constants (d_{3} geq0) and (d_{4}>0) such that biglvert g x,t bigrvert leq d_{3} + d_{4} vert tvert ^{beta-1}, for all (xinpartialOmega) and for all (tinBbb {R}), where (p
