Uniqueness of the large solution under radial symmetry

2020 ◽  
Vol 64 (10) ◽  
pp. 9-19
Author(s):  
V. V. Volchkov ◽  
Vit. V. Volchkov

2019 ◽  
Vol 6 (1) ◽  
pp. 13 ◽  
Author(s):  
Denise M. D. S. Mouga ◽  
Gabriel R. Schroeder ◽  
Nilton P. Vieira Junior ◽  
Enderlei Dec

The pollen morphology of thirteen species of Cactaceae was studied: M. backebergiana F.G. Buchenau, M. decipiens Scheidw, M. elongata DC, M. gracilis Pfeiff., M. hahniana Werderm., M. marksiana Krainz, M. matudae Bravo, M. nejapensis R.T. Craig & E.Y. Dawson, M. nivosa Link ex Pfeiff., M. plumosa F.A.C. Weber, M. prolifera (Mill.) Haw, M. spinosissima var. “A Peak” Lem. and M. voburnensis Scheer. All analysed pollen grains are monads, with radial symmetry, medium size (M. gracilis, M. marksiana, M. prolifera, large), tricolpates (dimorphs in M. plumosa [3-6 colpus] and M. prolifera [3-6 colpus]), with circular-subcircular amb (quadrangular in M. prolifera and M. plumosa with six colpus). The pollen grains presented differences in relation to the shape and exine thickness. The exine was microechinate and microperforated. The pollen morphological data are unpublished and will aid in studies that use pollen samples. These pollen grains indicate ornamental cacti.


Author(s):  
Riccardo Molle ◽  
Donato Passaseo

AbstractThe paper deals with the equation $$-\Delta u+a(x) u =|u|^{p-1}u $$ - Δ u + a ( x ) u = | u | p - 1 u , $$u \in H^1({\mathbb {R}}^N)$$ u ∈ H 1 ( R N ) , with $$N\ge 2$$ N ≥ 2 , $$p> 1,\ p< {N+2\over N-2}$$ p > 1 , p < N + 2 N - 2 if $$N\ge 3$$ N ≥ 3 , $$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$ a ∈ L loc N / 2 ( R N ) , $$\inf a> 0$$ inf a > 0 , $$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$ lim | x | → ∞ a ( x ) = a ∞ . Assuming that the potential a(x) satisfies $$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$ lim | x | → ∞ [ a ( x ) - a ∞ ] e η | x | = ∞ ∀ η > 0 , $$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$ lim ρ → ∞ sup a ( ρ θ 1 ) - a ( ρ θ 2 ) : θ 1 , θ 2 ∈ R N , | θ 1 | = | θ 2 | = 1 e η ~ ρ = 0 for some η ~ > 0 and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a(x) has radial symmetry, or that $$N=2$$ N = 2 , or that $$|a(x)-a_\infty |$$ | a ( x ) - a ∞ | is uniformly small in $${\mathbb {R}}^N$$ R N , etc. ....


2021 ◽  
pp. 1-8
Author(s):  
Junfeng Guo ◽  
Jian Han ◽  
Heyo Van Iten ◽  
Zuchen Song ◽  
Yaqin Qiang ◽  
...  

Abstract Hexangulaconulariids (Cambrian stages 1–2) are an extinct group of medusozoan polyps having a biradially symmetrical, fan-shaped periderm that is distinct from those of medusozoan polyps showing three-, four-, five-, or six-fold radial symmetry. Hexangulaconulariids exhibit substantial variation in gross morphology, including variation in the number of faces on each of the two major sides of the periderm. An intermediate taxon of hexangulaconulariids with ten faces (five on each major side) was expected. Here we describe a new hexangulaconulariid, Decimoconularia isofacialis new genus new species from Bed 5 of the Yanjiahe Formation (Cambrian Stage 2) in the Three Gorges area of Hubei Province, China. The new taxon differs from other hexangulaconulariids (Arthrochites, Hexaconularia, and Septuconularia) mainly in possessing a total of ten faces. The two lateral margins are each marked by a ridge in about the apertural half of the periderm and by a collinear furrow in about the apical half, while the five faces on each major side are bounded by a furrow in about the apertural half and by a collinear ridge in about the apical half. Among hexangulaconulariids, Decimoconularia and Septuconularia may be more closely related to each other than either genus is to Arthrochites or Hexaconularia. UUID: http://zoobank.org/ca270a3b-25ee-4d1f-bdeb-91a963370e70


2020 ◽  
Vol 268 (10) ◽  
pp. 5811-5822 ◽  
Author(s):  
Jinlu Li ◽  
Yanghai Yu ◽  
Weipeng Zhu

1968 ◽  
Vol 10 (1) ◽  
pp. 1-12 ◽  
Author(s):  
H. F. Black

Where a rotor runs within a clearance space, the clearance being comparable with rotor mass unbalance, the synchronous whirling behaviour of the rotor may be considerably affected by intermittent interaction with the stator at the clearance position. Discontinuity and jump phenomena may occur: in general, behaviour will be different with increasing speed from that with decreasing speed, and in either case zones may exist in which rotor-stator interaction is possible but not certain. In the analysis here presented, rotor and stator are regarded as linear multi-degree-of-freedom systems including damping; dry friction at the clearance space is taken into account. Discussion is limited to cases with radial symmetry, and interaction is assumed limited to the position of the clearance space. Polar receptances are used to establish equilibrium conditions with interaction, and speed zones are defined within which interaction may occur. Some hypothetical cases are fully explored, demonstrating that rotor-stator interactions may occur in a variety of forms and circumstances. Interactions with dry friction counterwhirling are also considered. Some experimental results on counterwhirl within a ball bearing are given and qualitatively compared with theory.


Author(s):  
Victor A. Galaktionov ◽  
Sergey A. Posashkov

SynopsisIn this paper we prove a certain monotonicity in time of non-negative classical solutions of the Cauchy problem for the quasilinear uniformly parabolic equation u1 = (ϕ(u))xx + Q(u) in wT = (0, T] × R1 with bounded sufficiently smooth initial function u(0, x) = uo(x)≧0 in Rl. We assume that ϕ(u) and Q(u) are smooth functions in [0, +∞) and ϕ′(u) >0, Q(u) > 0 for u > 0. Under some additional hypothesis on the growth of Q(u)ϕ′(u) at infinity, it is proved that if u(to, xo) becomes sufficiently large at some point (to, xo) ∈ wT, then ut(t, x0) ≧0 for all t ∈ [t0, T]. The proof is based on the method of intersection comparison of the solution with the set of the stationary solutions of the same equation. Some generalisations of this property for a quasilinear degenerate parabolic equation are discussed.


1963 ◽  
Vol 18 (8-9) ◽  
pp. 895-900
Author(s):  
Franz Peter Küpper

In a θ-pinch the radial symmetry of the electron density distribution as a function of time has been measured by a MACH—ZEHNDER interferometer. In a time interval of 400 nsec during a discharge an image converter made three pictures (exposure times of 10 nsec each) . Up to 100 nsec after the first compression, the experimental results show different density distributions for the cases of trapped parallel and antiparallel magnetic fields. Complete radial symmetry of the electron density distribution was not found.Another interferometric method for measuring the radial symmetry of the electron distribution by observing “zero order” fringes is described.


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