Asymptotic analysis of solutions to elliptic and parabolic problems

University dissertation from Matematiska institutionen

Abstract: In the thesis we consider two types of problems. In Paper 1, we studysmall solutions to a time-independent nonlinear elliptic partial differential equation of Emden-Fowler type in a semi-infnite cylinder. The asymptotic behaviour of these solutions at infnity is determined. First, the equation under the Neumann boundary condition is studied. We show that any solution small enough either vanishes at infnity or tends to a nonzero periodic solution to a nonlinear ordinary differential equation. Thereafter, the same equation under the Dirichlet boundary condition is studied, the non-linear term and right-hand side now being slightly more general than in the Neumann problem. Here, an estimate of the solution in terms of the right-hand side of the equation is given. If the equation is homogeneous, then every solution small enough tends to zero. Moreover, if the cross-section is star-shaped and the nonlinear term in the equation is subject to some additional constraints, then every bounded solution to the homogeneous Dirichlet problem vanishes at infnity.In Paper 2, we study asymptotics as t ? ? of solutions to a linear, parabolic system of equations with time-dependent coefficients in ?x(0,?), where ? is a bounded domain. On ??(0,?) we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function ?(t). This includes in particular situations when the coefficients may take different values on different parts of ? and the boundaries between them can move with t but stabilize as t ? ?. The main result is an asymptotic representation of solutions for large t. As a corollary, it is proved that if ??L1(0,?), then the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.