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Resumen:
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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) 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) infinity) u(t; u(0)) infinity) u(t; u(0)) 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.
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