Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term

Motivated by the work [9], in this paper we investigate the infinite boundary value problem associated with the semilinear PDE {Lu=f(u)+h(x)} on bounded smooth domains {\Omega\subseteq\mathbb{R}^{n}} , where L is a non-divergence structure uniformly elliptic operator with singular lower-order terms. In the equation, f is a continuous non-decreasing function that satisfies the Keller–Osserman condition, while h is a continuous function in Ω that may change sign, and which may be unbounded on Ω. Our purpose is two-fold. First we study some sufficient conditions on f and h that would ensure existence of boundary blow-up solutions of the above equation, in which we allow the lower-order coefficients to be singular on the boundary. The second objective is to provide sufficient conditions on f and h for the uniqueness of boundary blow-up solutions. However, to obtain uniqueness, we need the lower-order coefficients of L to be bounded in Ω, but we still allow h to be unbounded on Ω.

Multiple Positive Solutions for First-Order Impulsive Integrodifferential Equations on the Half Line in Banach Spaces

The author discusses the multiple positive solutions for an infinite boundary value problem of first-order impulsive superlinear integrodifferential equations on the half line in a Banach space by means of the fixed point theorem of cone expansion and compression with norm type.

Homoclinic solutions for linear and linearizable ordinary differential equations

Using functional arguments, some existence results for the infinite boundary value problemx˙=F(t,x),x(−∞)=x(+∞)are given. A solution of this problem is frequently called, from Poincaré, homoclinic.

Stability of the laminar boundary layer in a streamwise corner

This work examines the stability of viscous, incompressible flow along a streamwise corner, often called the corner boundary-layer problem. The semi-infinite boundary value problem satisfied by small-amplitude disturbances in the ‘blending boundary layer’ region is obtained. The mean secondary flow induced by the corner exhibits a flow reversal in this region. Uniformly valid ‘first approximations’ to solutions of the governing differ­ential equations are derived. Uniformity at infinity is achieved by a suitable choice of the large parameter and use of an appropriate Langer variable. Approximations to solutions of balanced type have a phase shift across the critical layer which is associated with instabilities in the case of two-dimensional boundary layer profiles.