Abstract:
Since 1960's, the blow-up phenomena for the Fujita type parabolic equation have been investigated by many researchers. In this survey paper, we discuss various results on the life span of positive solutions for several superlinear parabolic problems. In the last section, we introduce a recent result by the author. 1. Introduction 1.1. Fujita Type Results We first recall the result on the Cauchy problem for a semilinear heat equation: where , is the -dimensional Laplacian, and . Let be a bounded continuous function on . In pioneer work [1], Fujita showed that the exponent plays the crucial role for the existence and nonexistence of the solutions of (1.1). Let denote the Gaussian heat kernel: . Theorem 1.1 (see [1]). Suppose that and that its all derivatives are bounded.(i)Let . Then there is no global solution of (1.1) satisfying that？？ for？？ and .(ii)Let . Then for any there exists with the following property: if then there exists a global solution of (1.1) satisfying for and . In [2], Hayakawa showed first that there is no global solution of (1.1) in the critical case when or 2. Theorem 1.2 (see [2]). In case of , or , , (1.1) has no global solutions for any nontrivial initial data. In genaral space dimensions, Kobayashi et al. [3] consider the following problem: where and . Let be a bounded continuous function on . Theorem 1.3 (see [3]). Suppose that satisfies the following three conditions:(a) is a locally Lipschitz continuous and nondecreasing function in with？？ and for ,(b) for some ,(c)there exists a positive constant such that Then each positive solution of (1.5) blows up in finite time. Remark 1.4. (i) We remark that the proofs of the theorems in [2, 3] are mainly based on the iterated estimate from below obtained by the following integral equation: (ii) The critical nonlinearity of power type satisfies the assumptions (a), (b), and (c) in [3]. Weissler proved the nonexistence of global solution in -framework in [4]. The proof is quite short and elegant. Theorem 1.5 (see [4]). Suppose and that in ？ is not identically zero. Then there is no nonnegative global solution to the integral (1.7) with initial value . The outline of the proof is as follows. First we assume that there is a global solution. From the fact that the solution for some , we can obtain that . This contradicts the boundedness of for large . Hence the solution is not global. Existence and nonexistence results for time-global solutions of (1.1) are summarized as follows.(i)Let . Then every nontrivial solution of (1.1) blows up in finite time.(ii)Let . Then (1.1) has a time-global

Abstract:
By using the polycarbonate membrane a template, mesoporous silica rods are fabricated on a silicon substrate in one pot. From scanning electron microscope (SEM) images, the creation of fibrous morphology is confirmed over the entire area. The diameter of the obtained rods is consistent with that of the template. Transmission electron microscope (TEM) images revealed that the tubular mesochannels are uniaxially oriented parallel to the longitudinal axis of the silica rods. The mesoporous titania rods with anatase crystalline frameworks are also fabricated.

Abstract:
Silica and alumina with macro-meso-type hierarchical pore systems are synthesized by dual templating using both surfactants and polystyrene (PS) spheres. After calcination, scanning electron microscope images show uniform macropores with a diameter of approximately 200 nm. This size coincides with that of the original PS spheres. The density of the macropores increases with the amount of added PS spheres in the precursor solutions. Transmission electron microscope images, small-angle x-ray scattering spectra and N2 adsorption–desorption isotherms reveal the formation of ordered mesoporous structures in the macropore walls. Also, the existence of micropores (less than 2 nm in size) was confirmed from the large N2 uptake at low relative pressures.

Abstract:
We investigate the field-angle-dependent zero-energy density of states for YNi2B2C with using realistic Fermi surfaces obtained by band calculations. Both the 17th and 18th bands are taken into account. For calculating the oscillating density of states, we adopt the Kramer-Pesch approximation, which is found to improve accuracy in the oscillation amplitude. We show that superconducting gap structure determined by analyzing STM experiments is consistent with thermal transport and heat capacity measurements.

Abstract:
To determine the superconducting gap function of YNi2B2C, we calculate the local density of states (LDOS) around a single vortex core with the use of Eilenberger theory and the band structure calculated by local density approximation assuming various gap structures with point-nodes at different positions. We also calculate the angular-dependent heat capacity in the vortex state on the basis of the Doppler-Shift method. Comparing our results with the STM/STS experiment, the angular-dependent heat capacity and thermal conductivity, we propose the gap-structure of YNi2B2C, which has the point-nodes and gap minima along <110>. Our gap-structure is consistent with all results of angular-resolved experiments.

Abstract:
This paper investigates a variant of the Hamiltonian Cycle (HC) problem, named the Parity Hamiltonian Cycle (PHC) problem: The problem is to find a closed walk visiting each vertex odd number of times, instead of exactly once. We show that the PHC problem is in P even when a closed walk is allowed to use an edge at most z=4 times, by considering a T-join, which is a generalization of matching. On the other hand, the PHC problem is NP-complete when z=3. In the case of z=3 however, the problem is in P when an input graph is four-edge connected, but it still remains NP-complete even when it is two-edge connected. Thus, we are concerned with the hard case in detail, and give a simple necessary and sufficient condition that a two-edge connected C>=5-free (or P6-free) graph has a PHC. Note that the HC problem is known to be NP-complete for those graph classes. This subject is motivated by a new approach to connecting a hard problem HC and an easy problem T-join, by relaxing a constraint of HC.

Abstract:
We report the synthesis of hydroxyapatite nanoparticles (HANPs) by the coprecipitation method using calcium D-gluconate and potassium hydrogen phosphate as the sources of calcium and phosphate ions, respectively, and the triblock copolymer F127 as a stabilizer. The HANPs were characterized using scanning electron microscopy, x-ray diffraction, and nitrogen adsorption/desorption isotherms. Removal of F127 by solvent extraction or calcination alters the structure of HANPs. The solvent-extracted HANPs were single crystals with their lang001rang axis oriented along the rod axis of the HANP, whereas the calcined HANPs contained two crystal phases that resulted in a spherical morphology. The calcined HANPs had much higher surface area (127 m2 g 1) than the solvent-extracted HANPs (44 m2 g 1).

Abstract:
This review article summarizes recent developments in mesoporous titania materials, particularly in the fields of morphology control and applications. We first briefly introduce the history of mesoporous titania materials and then review several synthesis approaches. Currently, mesoporous titania nanoparticles (MTNs) have attracted much attention in various fields, such as medicine, catalysis, separation and optics. Compared with bulk mesoporous titania materials, which are above a micrometer in size, nanometer-sized MTNs have additional properties, such as fast mass transport, strong adhesion to substrates and good dispersion in solution. However, it has generally been known that the successful synthesis of MTNs is very difficult owing to the rapid hydrolysis of titanium-containing precursors and the crystallization of titania upon thermal treatment. Finally, we review four emerging fields including photocatalysis, photovoltaic devices, sensing and biomedical applications of mesoporous titania materials. Because of its high surface area, controlled porous structure, suitable morphology and semiconducting behavior, mesoporous titania is expected to be used in innovative applications.

Abstract:
We present the results of ammonia observations toward the center of NGC 3079. The NH3(J, K) = (1, 1) and (2, 2) inversion lines were detected in absorption with the Tsukuba 32-m telescope, and the NH3(1,1) through (6,6) lines with the VLA, although the profile of NH3(3,3) was in emission in contrast to the other transitions. The background continuum source, whose flux density was ~50 mJy, could not be resolved with the VLA beam of ~< 0."09 x 0."08. All ammonia absorption lines have two distinct velocity components: one is at the systemic velocity and the other is blueshifted, and both components are aligned along the nuclear jets. For the systemic components, the relatively low temperature gas is extended more than the high temperature gas. The blueshifted NH3(3,3) emission can be regarded as ammonia masers associated with shocks by strong winds probably from newly formed massive stars or supernova explosions in dense clouds in the nuclear megamaser disk. Using para-NH3(1,1), (2,2), (4,4) and (5,5) lines with VLA, we derived the rotational temperature Trot = 120 +- 12 K and 157 +- 19 K for the systemic and blueshifted components, respectively. The total column densities of NH3(0,0)-(6,6), assuming Tex ~Trot, were (8.85+-0.70) x 10^16 cm^-2 and (4.47+-0.78) x 10^16 cm-2 for the systemic and blueshifted components, respectively. The fractional abundance of NH3 relative to molecular hydrogen H2 for the systemic and blueshifted was [NH3]/[H2]=1.3x10^-7 and 6.5 x 10^-8, respectively. We also found the F = 4-4 and F = 5-5 doublet lines of OH 2{Pi}3/2 J = 9/2 in absorption, which could be fitted by two velocity components, systemic and redshifted components. The rotational temperature of OH was estimated to be Trot,OH >~ 175 K, tracing hot gas associated with the interaction of the fast nuclear outflow with dense molecular material around the nucleus.

Abstract:
Periodic mesoporous organosilica (PMO) spherical particles with different organic contents were synthesized in one pot by reacting 1,2-bis(triethoxysilyl)ethane (BTSE) with tetraethylorthosilicate (TEOS) using a spray-drying technique. The scanning electron microscopy observation of spray-dried products clearly showed the formation of spherical particles. The 29Si magic angle spinning nuclear magnetic resonance data revealed that the organic contents due to ethane fragments embedded in the frameworks were controllable and consistent with the BTSE/TEOS molar ratios of precursor solutions. Transmission electron microscopy, small-angle x-ray scattering, and N2 adsorption data of PMO with controlled organic contents indicated that the ethane fragments were embedded in the frameworks with the formation of ordered mesostructures. PMO with a high organic content (BTSE/TEOS=0.50) only showed a hydrophobic property. According to the same procedure, benzene groups were also integrated to a similar degree in the frameworks by using 1,4-bis(triethoxysilyl)benzene.