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A time--local solution is constructed for the Cauchy problem of the $n$-dimensional Navier--Stokes equations when the initial velocity belongs to Besov spaces of nonpositive order. The space contains $L^\infty$ in some exponents, so our solution may not decay at space infinity. In order to use iteration scheme we have to establish the Hölder type inequality for estimating bilinear term by dividing the sum of Besov norm with respect to levels of frequency. Moreover, by regularizing effect our solutions belong to $L^\infty$ for any positive time.
We consider a 3--dimensional Cauchy problem for a parabolic equation where the diffusion matrix has two eigenvalues which diverge with order larger than 2 and one eigenvalue which diverges with order less than 2, with respect to $|x|$, as $|x|\to \infty$. Order 2 of divergence is the critical value below which uniqueness and above which non--uniqueness results are known to hold in the set of bounded functions. Hence we are in an intermediate case. However we prove a uniqueness result, in which the presence of first order terms is crucial. Shauder type estimates of solutions are given too. The problem is of interest in the study of plasma physics.
A locally uniform stabilization result of the solutions of a coupled system of Korteweg- de Vries equations in a bounded domain is established. The main novelty is that internally only a localized damping mechanism is considered.
A thermoelastic system with the Gurtin-Pipkin model for heat conduction is considered. If the control time is large enough, under suitable conditions on the thermal conductivity kernel, exact controllability with boundary controls is established for both the displacement and the temperature.
The Sturm-Liouville equation $-y'' + qy = \lambda ry,$ on $ [0,l],$ is considered subject to the boundary conditions \noindent $y(0)\cos\alpha(0) = y'(0)\sin\alpha(0),$ $y(l)\cos\alpha(l) = y'(l)\sin\alpha(l).$ We assume that $r$ is piecewise continuous with a variety of behaviours at $0, l$ and an interior turning point. We give asymptotic approximations for the eigenvalues $\lambda_n$ of the above boundary value problem in forms equivalent to $\lambda_n = an^2+bn+O(n^\tau)$, where $\tau < 1$.
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