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We consider initial value problem for semilinear damped wave equations in three space dimensions. We show the small data global existence for the problem without the spherically symmetric assumption and obtain the sharp lifespan of the solutions. This paper is devoted to a proof of the Takamura's conjecture in [5] on the lifespan of solutions.

This paper is devoted to a proof of the conjecture in Takamura [16] on the lower bound of the lifespan of solutions to semilinear wave equations in two space dimensions. The result is divided into two cases according to the total integral of the initial speed.2010 Mathematics Subject Classification. 35L71, 35A01, 35E15. Key words and phrases. semilinear wave equations, initial value problem, lifespan, two space dimensions.Dedicated to Professor Nakao Hayashi on the occasion of his sixties birthday.

We study the initial value problem for the generalized cubic double dispersion equation in one space dimension. We establish a nonlinear approximation result to our global solutions that was obtained in [6]. Moreover, we show that as time tends to infinity, the solution approaches the superposition of nonlinear diffusion waves which are given explicitly in terms of the self-similar solution of the viscous Burgers equation. The proof is based on the semigroup argument combined with the analysis of wave decomposition.

We report the unintentional incorporation of Al during the growth of molecular beam epitaxy using RF plasma source, driven by N2 gas flow. The concentrations of N, Al, O, and C within GaNAs/GaAs/AlAs structure are investigated by secondary ion mass spectrometry. In spite of the closed shutter of Al cell, we observe Al incorporation with a concentration up to 1×1018 cm-3 in GaNAs layer and characteristically in the bottom side GaAs. Its concentration is solely dependent on N2 gas flow rate. Remarkably, the operation of the RF plasma has no impact on that. C and O show their concentrations corresponding to the extrinsic Al. The complex interactions between those elements predict a possible origin of material deteriorations and difficulty for the precise doping control.

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This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one-dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis. KEYWORDS hyperbolic relaxation system, large time behavior, optimal decay estimates, self-similar solution( 1.2) where we define the nonlinear term that f (u) = 2 u 2 + 3! u 3 with | | < 1, ≠ 0 and ∈ R. The subscripts t and x stand for the partial derivatives with respect to t and x, respectively.The damped wave equations with a nonlinear convection term (1.1) are derived from a nonlinear hyperbolic relaxation system by using the Chapman-Enskog expansion (cf Chern 1 and Liu 2 ). A nonlinear hyperbolic relaxation system is one of important mathematical models that describe physical phenomena. For example, non-equilibrium gas dynamics, magnetohydrodynamics, viscoelasticity, and flood flow with friction are expressed by nonlinear hyperbolic relaxation systems. Recently, the dissipative Timoshenko system, which describes the vibration of the beam, and Euler-Maxwell system, which describes the plasma physics, are studied by a lot of researchers (cf previous studies 3-6 ). These two physical models are nonlinear hyperbolic systems, which have a weak dissipative structure called the regularity-loss structure, and it is difficult to get the stability property. To analyze such complicated systems, it is very important to construct the several properties from the basic equations like (1.1). When = 0 in (1.1), Orive and Zuazua 7 studied the global existence and the asymptotic profile of the solutions of (1.1) and (1.2) and provided the initial data u 0 ∈ H 1 (R) ∩ L 1 (R) and u 1 ∈ L 2 (R) ∩ L 1 (R). Ueda and Kawashima 8 succeeded to construct the solutions of (1.1) and (1.2) with ≠ 0, provided the initial data u 0 ∈ W 1,p (R) ∩ L 1 (R) and u 1 ∈ L p (R) ∩ L 1 (R) for 1 ≤ p ≤ ∞, and derived that the solution tends to the self-similar solution of the Burgers equation: 7760