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Constant solutions, ground-state solutions and radial terrace solutions Du, Yihong


In many applications, one is interested in the nonlinear parabolic problem ut −∆u=f(u)(x∈RN,t>0), u(x,0)=u0(x)(x∈RN), where u0 ∈ L∞(RN) is nonnegative and has compact support, f is a smooth function satisfying f(0) = 0. One wants to know how much of the long-time dynamics of this problem is determined by the corre- sponding elliptic problem −∆u=f(u), u≥0(x∈RN). We show that, if u(·,t) stays bounded in L∞(RN) for all t > 0, then as t → ∞, u(·,t) converges to a stationary solution in L∞loc(RN), provided that all the zeros of f(u) in [0,∞) are nondegenerate (i.e., f(u) = 0 and u ≥ 0 imply f′(u) ̸= 0). Moreover, this stationary solution is either a stable constant solution (hence a stable zero of f ), or a ground-state solution based on a stable zero of f (namely a solution v(x) of the elliptic problem which is radially symmetric about some point x0 ∈ RN , decreases in |x−x0|, and lim|x|→∞ v(x) is a stable zero of f). Thus, viewed in the space L∞loc(RN), the long-time behavior of u(·,t) is determined by two simplest types of solutions of the corresponding elliptic problem. Furthermore, we show that, viewed in the space L∞(RN), the long- time behavior of u(·, t) resembles that of a radial terrace solution v(|x|, t), whose limit as t → ∞ is determined completely by a propagating ter- race of 1-space dimension, which by definition is a set of traveling wave solutions {wi}ki=1, with each wi solving the elliptic equation −wzz + ciwz = f(w)(z ∈ R1), where ci is a certain constant, called the wave speed of wi. This talk is based on joint works with Peter Polacik and with Hiroshi Matano.

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