Rodríguez Bernal, AníbalCholewa, Jan W.2023-06-202023-06-2020090218-202510.1142/S0218202509004029https://hdl.handle.net/20.500.14352/49699We 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.engjournal articlehttp://www.worldscinet.com/m3as/m3as.shtmlopen access517.9Locally uniform spacesExtremal stationary solutionsNonlinear logistic reaction termsParabolic problemsStabilityAsymptotic behavior of solutionsAttractorsEcuaciones diferenciales1202.07 Ecuaciones en Diferencias