Navier-Stokes flows are found on R² that decay in time more rapidly than observed in general. The decay rate is determined in accordance with the order of symmetry with respect to the action of dihedral groups of orthogonal transformations. Contrary to the previous work [13], the basic existence result is proved with no restriction on the size of initial data. Our result extends that of [3] under different assumptions on the initial data. Unlike [3], the proofs are all carried out without using estimates for the associated vorticity transport equations.

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In this paper we give new results on complete abstract second order differential equations of elliptic type in UMD Banach spaces. Existence, uniqueness and maximal regularity of the strict solution are proved using the celebrated Dore-Venni Theorem. This work completes the ones studied in the framework of Hölder spaces, see [7] and [8].

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We study the global existence, uniqueness, and asymptotic behavior of solutions to the Cauchy problem for the semilinear dissipative wave equations: (∅+∂_t)u =|u|^α+1 in R^N × (0, ∞) with u|_t=0=εu₀ and ∂_tu|_t=0 = εu₁ for a small parameter ε > 0. Here, we do not assume any compactly support conditions on the initial data (u₀, u₁). When dimension N = 4, 5 and α is greater than a critical number 2/N which is often called Fujita's exponent, we solve the global in time solvability problem and we derive the sharp decay rates of L^p norm with p ≥ 1 of the solutions.

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A nonnegative nontrivial solution of the quasilinear elliptic differential equation (1.1) below on the entire space is obtained. The function φ(t)t in the principal part is nonhomogeneous and b(t)t has the critical Orlicz-Sobolev growth with respect to φ.

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We study the spectrum of the solution operator for a linear Volterra difference equation in a Banach space, and obtain the decomposition of the phase space, together with that of the variation of constants formula in the phase space. As an application, we establish the existence of almost periodic solutions for forced linear Volterra difference equations with an almost periodic forcing term.

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We prove existence of scaling limits of sequences of functions defined by the recursion relation w′_n+1(x) = –w_n(x)². which is a successive approximation to w′(x) = –w(x)², a simplest non-linear ordinary differential equation whose solutions have moving singularities. Namely, the sequence approaches the exact solution as n → ∞ in an asymptotically conformal way, w_n(x) $\\asymp$ q_n $\\bar w$(q_nx), for a sequence of numbers {q_n} and a function $\\bar w$. We also discuss implication of the results in terms of random sequential bisections of a rod.

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In this paper we consider nontrivial stationary solutions to Chafee-Infante problem and the corresponding linearized eigenvalue problem. We introduce a reduced equation to give explicit and simple-looking expressions of some eigenvalues and eigenfunctions to linearized problem.

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