The paper is concerned with conditions for the existence of almost periodic solutions of the following abstract functional differential equation $¥dot u$(t) = Au(t) + [${¥mathcal B}u$](t) + f(t), where A is a closed operator in a Banach space X, ${¥mathcal B}$ is a general bounded linear operator in the function space of all X-valued bounded and uniformly continuous functions that satisfies a so-called autonomous condition. We develop a general procedure to carry out the decomposition that does not need the well-posedness of the equations. The obtained conditions are of Massera type, which are stated in terms of spectral conditions of the operator ${¥mathcal A}$ + ${¥mathcal B}$ and the spectrum of f. Moreover, we give conditions for the equation not to have quasi-periodic solutions with different structures of spectrum. The obtained results extend previous ones.

View full abstract

We give a reformulation of the symmetry of the Painlevé equations P_V, P_IV, P_III and P_II, respectively. Based on these constructions, we make natural extensions to fourth-order analogues for each of the Painlevé equations P_V and P_III, respectively.

View full abstract

We study the drift-diffusion system arising in the semiconductor device simulation and the plasma physics. We consider the initial value problem for this system and derive the optimal L^p decay estimate of solutions by applying the time weighted L^p energy method. Furthermore, we show that the solutions approach the corresponding heat kernels as time tends to infinity. This asymptotic result is based on the optimal L^p - L^q decay estimate for the linearized problem.

View full abstract

In this paper we shall deal with hyperbolic operators whose principal symbols can be microlocally transformed to symbols depending only on the fiber variables by homogeneous canonical transformations. We call such operators "hyperbolic operators with nearly constant coefficient principal part." Operators with constant coefficient hyperbolic principal part and hyperbolic operators with involutive characteristics belong to this class of operators. We shall give a necessary and sufficient condition for the Cauchy problem to be C^∞ well-posed under some additional assumptions. Namely, we shall generalize "Levi condition" and prove that the generalized Levi condition is necessary and sufficient for the Cauchy problem to be C^∞ well-posed.

View full abstract

In this paper we deal with degenerate second-order differential operators in the framework of weighted continuous function spaces on a real interval. We show that, under particular boundary conditions, such operators generate positive strongly continuous semigroups which are the transition semigroups associated with suitable Markov processes. Finally an application to the Black-Scholes equation is also discussed.

View full abstract

The formal and analytic classification of integrable singular linear differential equations has been studied among others by R. Gérard and Y. Sibuya. We provide a simple proof of their main result, namely: For certain irregular systems in two variables there is no Stokes phenomenon, i.e. there is no difference between the formal and the analytic classification.

View full abstract

We establish existence results for positive solutions of the boundary value problem
$¥left¥{ ¥begin{array}{l} (q(t)¥phi (u^{¥prime }))^{¥prime }+¥lambda r(t)f(u)=0,¥;t¥in (a,b) ¥¥ u(a)=u(b)=0,¥end{array}¥right.$
where φ(x) = |x|^{p - 2} x, p > 1, q ∈ C[a,b], r ∈ L¹ (a,b), f is allowed to have negative values and become infinite at 0, and λ is a positive parameter.

View full abstract