TY - JOUR
T1 - Complicated dynamics in scalar semilinear parabolic equations in higher space dimension
AU - Poláčik, Peter
PY - 1991/2
Y1 - 1991/2
N2 - We study the dynamics of the boundary value problem ut - Lu = g(x, u, ▽u), x ε{lunate} Ω, (1) u |∂Ω = 0, (2) where L is a second order uniformly elliptic operator and Ω ⊂RN is diffeomorphic to the ball in RN, N≥2. The main result asserts that given any Ck-vector field V on RN+1 with V(0) = 0 one can adjust coefficients of L and the function g such that the corresponding problem (1), (2) has an N+ 1-dimensional invariant manifold through the equilibrium u ≡ 0 and the Taylor expansion at u ≡ 0 of the vector field representing the flow on this manifold coincides (in appropriate coordinates) with the Taylor expansion of V, up to k-th order terms. This result implies that a hyperbolic invariant N-torus can be found in (1), (2) (if L and g are appropriately chosen). This result also indicates that "chaotic dynamics" is likely to occur for some choices of L and g.
AB - We study the dynamics of the boundary value problem ut - Lu = g(x, u, ▽u), x ε{lunate} Ω, (1) u |∂Ω = 0, (2) where L is a second order uniformly elliptic operator and Ω ⊂RN is diffeomorphic to the ball in RN, N≥2. The main result asserts that given any Ck-vector field V on RN+1 with V(0) = 0 one can adjust coefficients of L and the function g such that the corresponding problem (1), (2) has an N+ 1-dimensional invariant manifold through the equilibrium u ≡ 0 and the Taylor expansion at u ≡ 0 of the vector field representing the flow on this manifold coincides (in appropriate coordinates) with the Taylor expansion of V, up to k-th order terms. This result implies that a hyperbolic invariant N-torus can be found in (1), (2) (if L and g are appropriately chosen). This result also indicates that "chaotic dynamics" is likely to occur for some choices of L and g.
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U2 - 10.1016/0022-0396(91)90121-O
DO - 10.1016/0022-0396(91)90121-O
M3 - Article
AN - SCOPUS:0000331882
SN - 0022-0396
VL - 89
SP - 244
EP - 271
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 2
ER -