36 research outputs found
Orbital compactness and asymptotic behaviour of nonlinear parabolic systems with functionals
No full textSynopsisWeakly coupled semilinear parabolic systems of the form with homogeneous boundary conditions are studied. The nonlinear function g: C([−r, 0] × Ω ℝn) → ℝn is assumed to be locally Lipschitz continuous with r≧0 a given real number and Ω ⊂ ℝm a bounded domain, , ut for t ≧ 0 is denned by ut (σ, ξ) = u(t + σ, ξ), − r ≦σ ≦ 0 ξ ∊Ω and A is a uniformly elliptic second order diagonal operator. Let u be a bounded classical solution. We first establish precompactness results for the orbit of u in several function spaces. Using these results and assuming that a Liapunov function V is known for the corresponding ordinary functional differential equation ż =g(zt), we then show under some general conditions that the limit set ω+ (as t→∞) of u consists of spatially homogeneous functions only. Moreover, ω+ is invariant with respect to z = g(z,) and V = 0 on ω+. The proof uses a Liapunov function for the full system whichis obtained from V via a simple construction (cf. (3.3)). The theory is illustrated with an example.</jats:p
Über die C<sup>2</sup>-Kompaktheit der Bahn von Lösungen semflinearer parabolischer Systeme
No full textSynopsisThe semilinear parabolic systemut+A(x, D)u=g(u) in (0, ∞) × Ω, Ω⊂ℝnbounded,u∈ ℝN, with homogeneous boundary conditionsB(x, D)u=0 on (0, ∞)×∂Ω is considered. The non-linearitygis assumed to be locally Lipschitz-continuous. It is shown that the orbit of a bounded regular solutionuis relatively compact in.</jats:p
On the asymptotic behavior of a semilinear functional differential equation in Banach space
No full textInvariant sets for strongly coupled reaction-diffusion systems under general boundary conditions
No full textThe spectral radius and Liapunov's theorem
No full textAbstractA familiar theorem of Liapunov pertaining to stability of complex matrices is proved anew and extended to bounded linear operators on a Hilbert space. The extension depends on an identity of Taussky which connects equations of the form x − axb = c with those of the form ux + xv + w = 0. Another ingredient in our method is the notion of abscissa of stability, s(u), which corresponds under Taussky's transformation to the spectral radius r(a). When these ideas are combined it is found that a sharpened and generalized form of Liapunov's theorem follows from elementary properties of geometric series
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RARE KAON DECAYS: IL BUONO, IL BRUTTO, IL CATTIVO.
No full textThe author briefly reviews recent progress in rare kaon decays, where he takes ''rare'' to mean those with {Beta} < {Omicron}(10{sup -7})
