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.

UR - http://www.scopus.com/inward/record.url?scp=0000331882&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000331882&partnerID=8YFLogxK

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 -