Abstract. The aim of this paper is to give an existence theorem for a semilinear equation of evolution in the case when the generator of semigroup of operators depends on time parameter. The paper is a generalization of [2]. Basing on the notion of a measure of noncompactness in Banach space, we prove the existence of mild solutions of the equation considered. Additionally, the applicability of...

We consider the curve straightening flow of inextensible, open, planar curves generated by the Kirchhoff bending energy. It can be considered as a model for the motion of elastic, inextensible rods in a high friction regime. We derive governing equations, namely a semilinear fourth order parabolic equation for the indicatrix and a second order elliptic equation for the Lagrange multiplier. We p...

A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the backward Euler, Crank–Nicolson, and discontinuous Galerkin dG(r) methods are addressed. For their full discretizations, we employ elliptic reconstructions...

In this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent −∆u = λu− αu + u −1, u > 0, in Ω, u = 0, on ∂Ω. where Ω ⊂ Rn, n ≥ 3 is a bounded C2-domain λ > λ1, 1 < p < 2∗ − 1 = n+2 n−2 and α > 0 is a bifurcation parameter. Brezis and Nirenberg [2] showed that a lower order (non-negative) perturbation can contribute t...