We study asymptotics as t -> infinity of solutions to a linear, parabolic system of equations with time-dependent coefficients in Omega x (0, infinity), where Omega is a bounded domain. On partial derivative Omega x (0, infinity) 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 kappa(t). This includes in particular situations when the coefficients may take different values on different parts of Omega and the boundaries between them can move with t but stabilize as t -> infinity. The main result is an asymptotic representation of solutions for large t. As a corollary, it is proved that if kappa epsilon L-1 (0, infinity), then the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.

We study asymptotics as t --> infinity of solutions to a linear, parabolic system of equations with time-dependent coefficients in Omega x (0,infinity), where Omega is a bounded domain. On partial derivative Omega x (0,infinity) 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 kappa(t). This includes in particular situations when the coefficients may take different values on different parts of Omega and the boundaries between them can move with t but stabilize as t --> infinity. The main result is an asymptotic representation of solutions for large t. A consequence is that for kappa is an element of L-1(0,infinity), the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.

We study small solutions of a nonlinear partial differential equation in a semi-infinite cylinder. The asymptotic behaviour of these solutions at infinity isdetermined. First, the equation under the Neumann boundary condition is studied. We show that any solution small enough either vanishes at infinity ortends to a nonzero periodic solution of a nonlinear ordinary differential equation. Thereafter, the same equation under the Dirichlet boundary condition is studied, but now the nonlinear term and right-hand side are 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 of the homogeneous Dirichlet problem vanishes at infinity. An estimate for the solution is given.

4.

Rand, Peter

Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.

In the thesis we consider two types of problems. In Paper 1, we study

small 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 Ωx(0,∞), where Ω is a bounded domain. On δΩ(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 κ(t). This includes in particular situations when the coefficients may take different values on different parts of Ω 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 κєL1(0,∞), then the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.