The paper addresses two open problems related to global existence of solutions with a "linear-like" behavior at infinity. For a class of second-order nonlinear differential equations, we establish global existence of solutions under milder assumption on the rate of decay of the coefficient. Furthermore, as opposed to results reported in the literature, we prove for another class of second-order nonlinear differential equations that the region of the initial data for the solutions with desired asymptotic behavior is unbounded and proper.

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The paper provides some theorems on complete instability of zero solution relative to a set for nonautonomous nonlinear equations with delay. The right-hand side of the equation is assumed to satisfy conditions, which provide standard existence-uniqueness-continuous dependence-continuation theory for the equation, as well as precompactness of the collection of translations in time of the right-hand side in a functional space with a metrizable compact open topology. These assumptions allow constructing limiting equations. Using conceptions of Lyapunov-Razumikhin functions and limiting equations, new instability results are obtained, which are applicable, in particular, to autonomous and periodic delayed differential equations and generalize some known results.

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In this work we give some new results on complete abstract second order differential equations of elliptic type in a Banach space. The existence and the uniqueness of the strict solution are proved under some natural assumptions generalising previous theorems on the subject.

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This paper is concerned with the Boussinesq equations which describe the heat convection in a viscous incompressible fluid. Local existence and uniqueness theorems are established for the n-dimensional Boussinesq equations in the whole space with nondecaying initial data. In two dimensional case the solution can be extended globally in time without smallness of the initial data.

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This paper is concerned with the initial value problem for some diffusion system which describes the process of a pattern formation of biological individuals by chemotaixis and growth. In the paper Osaki et al. [13], exponential attractors have been constructed for the dynamical system determined by this problem. The exponential attractor is one of limit sets which is a positively invariant compact set with finite fractal dimension and which attracts every trajectory in an exponential rate. In this paper we study another feature of exponential attractors, that is we show that the approximate solution also gets close to the exponential attractor in an exponential rate and remains in its neighborhood forever. Our methods are available to any other exponential attractors determined by interaction-diffusion systems.

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R. Ikehata recently proved some integral estimate for the difference between the solution of an abstract heat equation and the solution of an abstract wave equation which results from the heat equation by a time singular perturbation. The estimate is obtained if the initial values are chosen appropriately. We prove a pointwise estimate which improves the above result for large times into several directions, and we also establish the optimality of this estimate for the wave equation in an exterior domain. Our proofs rely on the spectral theorem for unbounded self-adjoint operators.

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We present a new expression for the elliptic-difference Painlevé equation. As in our construction all parameters in their equations appear in a symmetric way, the permutation symmetry of the equations is immediately apparent. We present expressions for other discrete Painlevé equations, obtained in a similar way.

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The paper is concerned with conditions for all mild solutions of abstract functional differential equations with finite delay in a Banach space to be periodic and asymptotic periodic, where forcing term is a continuous 1-periodic function. The obtained results extend various recent ones on the subject.

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This paper studies the initial value problem of Boussinesq-type system which describes the motion of water waves. We show the time local well-posedness in the weighted Sobolev space. This is the generalization of Angulo's work [1] from the view of regularity. Our argument is based on the contraction mapping principle for the integral equations after reducing our problem into the derivative nonlinear Schrödinger system. To overcome the regularity loss in the nonlinearity, we shall apply the smoothing effects of linear Schrödinger group due to Kenig-Ponce-Vega [7]. The gauge transform is also used to remove size restriction on the initial data.

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It will be proved that Painlevé first transcendents and their first derivatives are algebraically independent over the rational function field with complex coefficients, by the use of the irreducibility. A particular case indicates that the group of differential automorphisms of the differential field generated by a Painlevé first transcendent over the rational function field is trivial.

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