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We consider a reaction–diffusion–advection equation of the form: for , where is a -periodic function, is a -periodic Fisher–KPP type of nonlinearity with changing sign, is a free boundary satisfying the Stefan condition. We study the long time behavior of solutions and find that there are two critical numbers and with , and , such that a vanishing–spreading dichotomy result holds when ; a vanishing–transition–virtual spreading trichotomy result holds when ; all solutions vanish when or . 相似文献
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We study a non-standard infinite horizon, infinite dimensional linear–quadratic control problem arising in the physics of non-stationary states (see e.g. Bertini et al. (2004, 2005)): finding the minimum energy to drive a given stationary state (at time ) into an arbitrary non-stationary state (at time ). This is the opposite to what is commonly studied in the literature on null controllability (where one drives a generic state into the equilibrium state ). Consequently, the Algebraic Riccati Equation (ARE) associated with this problem is non-standard since the sign of the linear part is opposite to the usual one and since its solution is intrinsically unbounded. Hence the standard theory of AREs does not apply. The analogous finite horizon problem has been studied in the companion paper (Acquistapace and Gozzi, 2017). Here, similarly to such paper, we prove that the linear selfadjoint operator associated with the value function is a solution of the above mentioned ARE. Moreover, differently to Acquistapace and Gozzi (2017), we prove that such solution is the maximal one. The first main result (Theorem 5.8) is proved by approximating the problem with suitable auxiliary finite horizon problems (which are different from the one studied in Acquistapace and Gozzi (2017)). Finally in the special case where the involved operators commute we characterize all solutions of the ARE (Theorem 6.5) and we apply this to the Landau–Ginzburg model. 相似文献
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《Expositiones Mathematicae》2022,40(4):1049-1083
We summarize known criteria for the non-existence, existence and on the number of limit cycles of autonomous real planar polynomial differential systems, and also provide new results. We give examples of systems which realize the maximum number of limit cycles provided by each criterion. In particular we consider the class of differential systems of the form where are natural numbers with and for , are quasi-homogeneous vector fields. 相似文献
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《Discrete Mathematics》2020,343(10):111996
A Gallai coloring of a complete graph is an edge coloring without triangles colored with three different colors. A sequence of positive integers is an -sequence if . An -sequence is a G-sequence if there is a Gallai coloring of with colors such that there are edges of color for all . Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer there exists an integer such that every -sequence is a G-sequence if and only if . They showed that and .We show that and give almost matching lower and upper bounds for by showing that with suitable constants , for all sufficiently large . 相似文献
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Yinan Guo 《Expositiones Mathematicae》2021,39(2):165-181
Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set . It is shown that, for each , every real number in the unit interval is the sum with each in and some . Furthermore, every real number in the interval can be written as , the sum of eight cubic powers with each in . Another Cantor set is also considered. More specifically, when is embedded into the complex plane , the Waring–Hilbert problem on has a positive answer for powers less than or equal to 4. 相似文献
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In this paper, we study the long-time behavior of solutions of a reaction–diffusion model in a one-dimensional river network, where the river network has two branches, and the water flow speeds in each branch are the same constant . We show the existence of two critical values and 2 with , and prove that when , the population density in every branch of the river goes to 1 as time goes to infinity; when , then, as time goes to infinity, the population density in every river branch converges to a positive steady state strictly below 1; when , the species will be washed down the stream, and so locally the population density converges to 0. Our result indicates that only if the water-flow speed is suitably small (i.e., ), the species will survive in the long run. 相似文献
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In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over with designed distance for all , where q is a prime power and is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16], and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range for , where if m is odd, and if m is even. 相似文献
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We look for positive solutions for the singular equation where , , is a parameter, and has some summability properties. By using a perturbation method and critical point theory, we obtain two solutions when and the parameter is small. 相似文献
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Minimal blocking sets in have size at most . This result is due to Bruen and Thas and the bound is sharp, sets attaining this bound are called unitals. In this paper, we show that the second largest minimal blocking sets have size at most , if , , or , , . Our proof also works for sets having at least one tangent at each of its points (that is, for tangency sets). 相似文献
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In Korchmáros et al. (2018)one-factorizations of the complete graph are constructed for with any odd prime power such that either or . The arithmetic restriction is due to the fact that the vertices of in the construction are the points of a conic in the finite plane of order . Here we work on the Euclidean plane and describe an analogous construction where the role of is taken by a regular -gon. This allows us to remove the above constraints and construct one-factorizations of for every even . 相似文献
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《Annals of Pure and Applied Logic》2022,173(8):103135
We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable P-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which , each of , , has a witness and there is a well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a well-order of the reals is consistent with and each of the following: , , , where the smaller cardinal characteristics have co-analytic witnesses.Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type. 相似文献