TY - JOUR

T1 - Localized solutions of a semilinear parabolic equation with a recurrent nonstationary asymptotics

AU - Poláčik, Peter

AU - Yanagida, Eiji

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We examine the behavior of positive bounded, localized solutions of semilinear parabolic equations ut = Δu + f(u) on ℝN. Here f ∈ C1, f(0) = 0, and a localized solution refers to a solution u(x, t) which decays to 0 as x→∞ uniformly with respect to t > 0. In all previously known examples, bounded, localized solutions are convergent or at least quasi-convergent in the sense that all their limit profiles as t→∞are steady states. If N = 1, then all positive bounded, localized solutions are quasi-convergent. We show that such a general conclusion is not valid if N ≥ 3, even if the solutions in question are radially symmetric. Specifically, we give examples of positive bounded, localized solutions whose ω-limit set is infinite and contains only one equilibrium.

AB - We examine the behavior of positive bounded, localized solutions of semilinear parabolic equations ut = Δu + f(u) on ℝN. Here f ∈ C1, f(0) = 0, and a localized solution refers to a solution u(x, t) which decays to 0 as x→∞ uniformly with respect to t > 0. In all previously known examples, bounded, localized solutions are convergent or at least quasi-convergent in the sense that all their limit profiles as t→∞are steady states. If N = 1, then all positive bounded, localized solutions are quasi-convergent. We show that such a general conclusion is not valid if N ≥ 3, even if the solutions in question are radially symmetric. Specifically, we give examples of positive bounded, localized solutions whose ω-limit set is infinite and contains only one equilibrium.

KW - Asymptotic behavior

KW - Localized solutions

KW - Nonconvergent solutions

KW - Semilinear parabolic equation

UR - http://www.scopus.com/inward/record.url?scp=84910144979&partnerID=8YFLogxK

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U2 - 10.1137/140958566

DO - 10.1137/140958566

M3 - Article

AN - SCOPUS:84910144979

VL - 46

SP - 3481

EP - 3496

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 5

ER -