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In this paper, we consider the Cauchy problem for a two-phase model with magnetic field in three dimensions. The global existence and uniqueness of strong solution as well as the time decay estimates in are obtained by introducing a new linearized system with respect to for constants and , and doing some new a priori estimates in Sobolev Spaces to get the uniform upper bound of in norm. 相似文献
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Tristan Roy 《Journal of Differential Equations》2018,264(9):6013-6024
The purpose of this corrigendum is to point out some errors that appear in [1]. Our main result remains valid, i.e scattering of solutions of the loglog energy-supercritical Schrödinger equation , , , with , radial data but with slightly different values of , i.e if and if . We propose some corrections. 相似文献
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Let be the -dimensional complex projective space, and let be two non-empty open subsets of in the Zariski topology. A hypersurface in induces a bipartite graph as follows: the partite sets of are and , and the edge set is defined by if and only if . Motivated by the Turán problem for bipartite graphs, we say that is -grid-free provided that contains no complete bipartite subgraph that has vertices in and vertices in . We conjecture that every -grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in is bounded by a constant , and we discuss possible notions of the equivalence.We establish the result that if is -grid-free, then there exists of degree in such that . Finally, we transfer the result to algebraically closed fields of large characteristic. 相似文献
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In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): , , and (r20): , , and the periodic orbits of the quadratic isochronous centers , , and , . The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system and are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line . It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively and counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers and are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively. 相似文献
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