Gradient-like parabolic semiflows on \(BUC(\mathbb R^N)\)

Daniel Daners and Sandro Merino
Proceedings of the Royal Society of Edinburgh 128A(1998), 1281–1291
Original available at doi:S0308210500027323
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We prove that a class of weighted semilinear reaction diffusion equations on \(\mathbb R^N\) generates gradient-like semiflows on the Banach space of bounded uniformly continuous functions on \(\mathbb R^N\). If \(N = 1\) we show convergence to a single equilibrium. The key for getting the result is to show the exponential decay of the stationary solutions, which is obtained by means of a decay estimate of the kernel of the underlying semigroup.

AMS Subject Classification (2000): 47D06, 35B09, 35C10

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