Extremal equilibria for dissipative parabolic equations in locally uniform spaces

Abstract

We consider a reaction diffusion equation u(t) = Delta u + f(x, u) in R-N with initial data in the locally uniform space (L) over dot(U)(q)(R-N), q is an element of [1, infinity), and with dissipative nonlinearities satisfying sf(x, s) <= C(x)s(2) + D(x)vertical bar s vertical bar, where C is an element of L-U(r1)(R-N) and 0 <= D is an element of L-U(r2)(R-N) for certain r(1), r(2) > N/2. We construct a global attractor A and show that A is actually contained in an ordered interval [phi(m), phi(M)], where phi(m), phi(M) is an element of A is a pair of stationary solutions, minimal and maximal respectively, that satisfy phi(m) <= lim inf(t ->infinity) u(t; u(0)) <= lim sup(t ->infinity) u(t; u(0)) <= phi(M) uniformly for u(0) in bounded subsets of (L) over dot(U)(q)(R-N). A sufficient condition concerning the existence of minimal positive steady state, asymptotically stable from below, is given. Certain sufficient conditions are also discussed ensuring the solutions to be asymptotically small as vertical bar x vertical bar ->infinity. In this case the solutions are shown to enter, asymptotically, Lebesgue spaces of integrable functions in R-N, the attractor attracts in the uniform convergence topology in RN and is a bounded subset of W-2,W-r (R-N) for some r > N/2. Uniqueness and asymptotic stability of positive solutions are also discussed.