PUB - Publikationen an der Universität Bielefeld

One studies here, via the invariance principle for nonlinear semigroups in Banach spaces [8], the properties of the omega-limit set omega(u(0)) corresponding to the orbit gamma(u(0)) ={u(t, u(0)); t >= 0}, where u = u(t, u(0)) is the solution to the nonlinear Fokker-Planck equation u(t) - Delta beta(u) + div(Db(u)u) = 0 in (0,infinity) x R-d, u(0,x) = u(0)(x), x is an element of R-d, u(0) is an element of L-1 (R-d), d >= 3. Here, ss is an element of C-1( R) and ss '(r) > 0, for all r not equal 0. Moreover, ss is a sublinear function, possibly degenerate in the origin, b is an element of C-1(R), bbounded, b >= b(0) is an element of(0, infinity), D is bounded such that D= -del Phi, where Phi is an element of C(R-d) is such that Phi >= 1, Phi(x) -> infinity as vertical bar x vertical bar -> infinity and satisfies a condition of the form Delta Phi - alpha vertical bar del Phi vertical bar(2) <= 0, a.e. on R-d. The main conclusion is that the equation has an equilibrium state and the set omega(u(0)) is a nonempty, compact subset of L-1(R-d) while, for each t >= 0, the operator u(0) -> u(t, u(0)) is an isometry on omega(u(0)). In the nondegenerate case 0 < gamma(0) <= ss ' <= gamma(1) studied in [2], it follows that lim (t ->infinity) S(t)u(0) = u(infinity) in L-1(R-d), where u(infinity) is the unique bounded stationary solution to the equation. (c) 2022 Elsevier Inc. All rights reserved.