Existence,Non-existence,and Uniqueness for a Heat Equation with Exponential Nonlinearity in ℝ2

We consider a semilinear heat equation with exponential nonlinearity in ?². We prove that local solutions do not exist for certain data in the Orlicz space exp L ²(?²), even though a small data global existence result holds in the same space exp L ²(?²). Moreover, some suitable subclass of exp L ²(?²) for local existence and uniqueness is proposed.

     

Sharp remainder of a critical Hardy inequality

An explicit representation is given to the remainder of a critical Hardy inequality in \({L^{n}(\mathbb{R}^{n})}\) with \({n \ge 2}\).     

Critical dissipative estimate for a heat semigroup with a quadratic singular potential and critical exponent for nonlinear heat equations

We consider a heat semigroup with an inverse square potential and prove critical dissipative estimates including the endpoint case. Some application to a nonlinear heat equation with an inverse square potential is also discussed.     

Large-Scale Production of CdSe Nanocrystal by a Continuous Flow Reactor

Organically capped CdSe nanocrystals were successfully produced by a continuous flow reactor in 13 g/h rate as isolated CdSe nanocrystal, using trioctylphosphine oxide (TOPO) both as the capping organic reagent and the high-temperature reaction solvent. Relatively high reaction temperature (e.g. 350°C) was necessary for matured crystal growth. The quality of TOPO (i.e. impurity composition such like phosphonic acids) was also influential on the quality of the resulting CdSe nanocrystal. The continuous flow reactor was able to produce highly-luminescence, monodispersed CdSe nanocrystals, confirmed by transmission electron microscope observation. The production rate was stable at least 1 h to allow over 10 g production.     

Adhesion Behavior of Microorganisms Isolated from Soil on Hydroxyapatite and Other Materials

Applied Biochemistry and Biotechnology - The adhesion behavior of microorganisms on different materials was examined to obtain basic knowledge for designing support materials for microorganisms....     

Non-uniqueness for a critical heat equation in two dimensions with singular data

Nonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function $\stackrel{?}{u}$ the problem is well-posed, b) for initial data above this threshold function $\stackrel{?}{u}$, there exists no solution, c) for the singular initial datum $\stackrel{?}{u}$ there is non-uniqueness. The function $\stackrel{?}{u}$ is a weak stationary singular solution of the problem, and we show that there exists also a regularizing classical solution with the same initial datum $\stackrel{?}{u}$.     

Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation

We exhibit the optimal constant for Sobolev inequalities in Lorentz spaces for a mean oscillation, and its relation with a boundedness of the Hardy–Littlewood maximal operator in Sobolev spaces. Some applications to a scale invariant form of Hardy?s inequality in a limiting case are also considered.