A Remark on the Existence of Suitable Vector Fields Related to the Dynamics of Scalar Semilinear Parabolic Equations

(1) ut = u+ f (x; u;ru) ; x 2 ; t > 0; @u @ = 0 on @ ; on suitable Sobolev spaces has attracted much interest (see [A, H] and more recent references at the end of this note). Many e¤orts have been made to show the complexity of its dynamical behavior (see some survey papers and recent articles [DP, P1, P2, P3, Pr, PR, R] and the references therein). In particular, the following nice result was proven in [P2]: if there exists a smooth vector …eld on , = ( 1; ; n) such that rank ( (x) ; @1 (x) ; ; @n (x)) = n for all x 2 ; @ @ = 0 on @ , then for any smooth vector …eld X on R, there exists a smooth function f , such that span f 1; ; ng is invariant under (1) and for any integral curve of X, c = c (t), u = Pn i=1 ci (t) i (x) is a solution to (1). Moreover, it was shown that such kind of vector …eld always exists on a starshaped domain. The main result of this short note is a classi…cation of all the domains on which one may …nd this type of vector …elds. More precisely, we have