The PDE ∆^m u = u|u|^(p-1), with m ∈ N and p ∈ (1,∞), serves as a paradigm for a large class of higher-order elliptic equations with power-like nonlinearities. In studying radially symmetric solutions of this PDE and their asymptotic behavior, it is of critical importance to understand the dynamics of the associated ODE, u^(2m)= u|u|^(p-1), which is the subject of the present paper. Most solutions of the ODE blow up in finite time, diverging to either ∞ or −∞. In the phase space R^(2m), the orbits of these two types of solutions are separated by a (2m−1)-dimensional manifold M, which is unordered and homeomorphic to each of the coordinate hyperplanes under orthogonal projection. Solutions with orbits on M do not blow up or,else,are unbounded from above and from below (oscillatory blow-up). In the second-order case, M coincides with the stable manifold of the trivial equilibrium. In the fourth-order case, which is our main focus, M contains a two-dimensional manifold M_0 of periodic orbits. There exists a unique periodic solution ũ with ũ (0)=1 and ũ’(0)=0, which shares most of the symmetry properties of the common cosine; all nontrivial periodic solutions are obtained from ũ via scaling and phase-shifting. Every solution with orbit on M \M_0 converges to either the trivial equilibrium or one of the nontrivial periodic orbits; oscillatory blow-up does not occur. In higher-order cases, the structure of the manifold M remains largely open.
Periodic Solutions and Invariant Manifolds for an Even-Order Differential Equation with Power Nonlinearity
LAZZO, Monica;
2011-01-01
Abstract
The PDE ∆^m u = u|u|^(p-1), with m ∈ N and p ∈ (1,∞), serves as a paradigm for a large class of higher-order elliptic equations with power-like nonlinearities. In studying radially symmetric solutions of this PDE and their asymptotic behavior, it is of critical importance to understand the dynamics of the associated ODE, u^(2m)= u|u|^(p-1), which is the subject of the present paper. Most solutions of the ODE blow up in finite time, diverging to either ∞ or −∞. In the phase space R^(2m), the orbits of these two types of solutions are separated by a (2m−1)-dimensional manifold M, which is unordered and homeomorphic to each of the coordinate hyperplanes under orthogonal projection. Solutions with orbits on M do not blow up or,else,are unbounded from above and from below (oscillatory blow-up). In the second-order case, M coincides with the stable manifold of the trivial equilibrium. In the fourth-order case, which is our main focus, M contains a two-dimensional manifold M_0 of periodic orbits. There exists a unique periodic solution ũ with ũ (0)=1 and ũ’(0)=0, which shares most of the symmetry properties of the common cosine; all nontrivial periodic solutions are obtained from ũ via scaling and phase-shifting. Every solution with orbit on M \M_0 converges to either the trivial equilibrium or one of the nontrivial periodic orbits; oscillatory blow-up does not occur. In higher-order cases, the structure of the manifold M remains largely open.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
