The generic gradient-like structure of certain asymptotically autonomous\n semilinear parabolic equations

2017/12/13 by Axel Jänig, Jänig, Axel
Computer Science · Mathematics · #35B40 #35K58 #37D05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1712.04838

openalex publication_date 2017/12/13 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We consider asymptotically autonomous semilinear parabolic equations u_t + Au\n= f(t,u). Suppose that f(t,.)\→ f^\± as t\→\±\∞, where the\nsemiflows induced by u_t + Au = f^\±(u) are\ngradient-like. Under certain assumptions, it is shown that generically with\nrespect to a perturbation g with g(t)\→ 0 as |t|\→\∞, every\nsolution of u_t + Au = f(t,u) + g(t) is a connection between equilibria e^\±\nof eqrefeq:140602-1511 with m(e^-)\≥ m(e⁺). Moreover, if the Morse\nindices satisfy m(e^-) = m(e⁺), then u is isolated by linearization.\n

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