In this paper, some properties of three-dimensional Stokes equations with boundary condition of pressure on parts of boundary are studied. By use of these properties, approximate controllability by tangent boundary controls acting on a subboundary is studied. In addition, the controllability problem is considered when its controls act on a subdomain.

In this paper a class of p-Laplace type elliptic equations with unbounded coefficients on R^N is considered. It is proved that there exist radial solutions on R^N. On sufficiently large ball, radial and nonradial solutions are obtained. Finally, some necessary conditions for the existence of solutions are given.

On the assumption that the total variation of the initial data is suffi- ciently small, we can use the stability results of Dafermos to get the L2 estimate of the difference between the solutions to the isentropic steady Euler system and the potential flow equations with the same initial data.

In this note we consider the blow-up rate of solutions for p-Laplacian equation with nonlinear source, u_t=div(|∇u|^{p-2}∇u)+u^q, (x,t)∈R^N×(0,T), N ≥ 1. When q > p - 1, the blow-up rate of solutions is studied.

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R_+ u_{tt}-u_{xx}+u_t+f(u)_x=0, t > 0, x ∈ R_+, u(0,x)=u_0(x)→ u_+, as x→+∞, u_t(0,x)=u_1(x), u(t,0)=u_b. For the non-degenerate case f'(u_+) < 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u(t, x) which converges to the stationary wave φ(x) uniformly in x ∈ R+ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates (including the algebraic convergence rate and the exponential convergence rate) of u(t, x) toward φ(x) are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f'(u_b) < 0. The main purpose of this paper is devoted to discussing the case of f'(u_b) = 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.

We consider the asymptotic behavior of solutions to a model of hyperbolic- elliptic coupled system on the half-line R_+ = (0,∞), u_t+uu_x+q_x=0, -q_{xx}+q+u_x=0, with the Dirichlet boundary condition u(0, t) = 0. S. Kawashima and Y. Tanaka [Kyushu J. Math., 58(2004), 211-250] have shown that the solution to the correspond- ing Cauchy problem behaviors like rarefaction waves and obtained its convergence rate when u_-< u_+. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, asymptotic behavior of the solution (u, q) is proved to be rarefaction wave as t tends to infinity. Its convergence rate is also obtained by the standard L^2-energy method and L^1-estimate. It decays much lower than that of the corresponding Cauchy problem.