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This paper is concerned with initial-boundary-value problems (IBVPs) for a class of nonlinear Schrödinger equations posed either on a half line R + or on a bounded interval ( 0 , L ) with nonhomogeneous boundary conditions. For any s with 0 ≤ s < 5 / 2 and s ≠ 1 / 2 , it is shown that the relevant IBVPs are locally well-posed if the initial data lie in the L 2 -based Sobolev spaces H s ( R + ) in the case of the half line and in H s ( 0 , L ) on a bounded interval, provided the boundary data are selected from H l o c ( 2 s + 1 ) / 4 ( R + ) and H ( s + 1 ) / 2 l o c ( R + ) , respectively. (For s > 1 2 , compatibility between the initial and boundary conditions is also needed.) Global well-posedness is also discussed when s ≥ 1 . From the point of view of the well-posedness theory, the results obtained reveal a significant difference between the IBVP posed on R + and the IBVP posed on ( 0 , L ) . The former is reminiscent of the theory for the pure initial-value problem (IVP) for these Schrödinger equations posed on the whole line R while the theory on a bounded interval looks more like that of the pure IVP posed on a periodic domain. In particular, the regularity demanded of the boundary data for the IBVP on R + is consistent with the temporal trace results that obtain for solutions of the pure IVP on R , while the slightly higher regularity of boundary data for the IBVP on ( 0 , L ) resembles what is found for temporal traces of spatially periodic solutions. [ABSTRACT FROM AUTHOR]

CCR10, a member of the chemokine receptor subfamily, is overexpressed in several tumors and play a crucial role in cancer development and progression. However, the functions of CCR10 in breast cancer are unknown. Here, we detected the protein levels of CCR10 in breast cancer cells by western blotting, and examined CCR10 expression in breast cancer tissues via immunohistochemical assay. The results showed that CCR10 expression was elevated in breast cancer MCF7, BT-474 and MDA-MB-231 cells. Further, 63 of 89 cases (70.8%) had positive CCR10 staining, and the CCR10 level was closely related to capsular invasion, lymph node metastasis and tumor stage. Moreover, CCL27, the ligand of CCR10, dose-dependently stimulated the invasion and migration of breast cancer cells, and promoted MMP-7 expression and ERK1/2 activation. CCR10 knockdown in breast cancer cells through siRNA transfection attenuated CCL27-induced cell invasiveness, and suppressed MMP-7 expression and ERK1/2 activation. Additionally, blocking the ERK1/2 pathway inhibited the CCL27/CCR10-promoted cell invasion of breast cancer cells. Together, these data suggest that CCL27/CCR10 interaction induces the ERK1/2 pathway, which then increases MMP-7 expresion and subsequently promotes breast cancer cell invasion and migration. Thus, CCR10 may be a key regulator in breast cancer cell invasion and migration. [ABSTRACT FROM AUTHOR]

Considered in this paper is the initial boundary value problem (IBVP) of the Kawahara equation, a class of the fifth order KdV equation, posed on a finite interval, (0.1) u t + u x + β u x x x + u x x x x x + u u x = 0 , 0 < x < L , t > 0 , u (x , 0) = ϕ (x) , subject to the non-homogeneous boundary conditions (0.2) B j u = h j (t) , j = 1 , 2 , 3 , 4 , 5 t > 0 where B j u = ∑ k = 0 4 (a j k ∂ x k u (0 , t) + b j k ∂ x k u (L , t)) , j = 1 , 2 , 3 , 4 , 5 and a j k , b j k (k = 0 , 1 , 2 , 3 , 4 and j = 1 , 2 , 3 , 4 , 5) are real constants. Under some general assumptions imposed on the coefficients a j k , b j k , the IBVP (0.1)–(0.2) is shown to be locally well posed in the space H s (0 , L) for any s ≥ 0 with naturally compatible ϕ ∈ H s (0 , L) and boundary values h j , j = 1 , 2 , 3 , 4 , 5 belonging to some appropriate spaces with optimal regularity. The sharp Kato smoothing properties (due to Kenig, Ponce and Vega (Kenig et al., 1991; Kenig et al., 1993)) of the pure initial value problem (IVP) of the linear inhomogeneous fifth order KdV equation posed on the whole line R , v t + β v x x x + v x x x x x = g (x , t) , v (x , 0) = ψ (x) , x , t ∈ R , have played an important role in establishing the well-posedness of the IBVP (0.1)–(0.2). [ABSTRACT FROM AUTHOR]