We give a new proof of temporally global existence of small solutions for systems of semi-linear wave equations. Our proof uses the Klainerman-Sideris inequality and a space-time L²-estimate. We also discuss whether the scale-invariant version of the space-time L²-estimates can hold, and obtain some related estimates. Among other things, we prove that the Keel-Smith-Sogge estimate actually holds in all space dimensions.

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We state sufficient conditions for a certain nonlinear planar differential equation to have only oscillatory solutions. We also obtain a result of this type for a case where the equation is subjected to an impulsive condition.

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The stability of the null solution of the equation (1.1) below is investigated. The main arguments are based on some Bernoulli type differential inequalities. Extensions to the whole real line are also discussed.

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The generalized hypergeometric differential equations of rank n whose monodromy groups are the symmetric group of degree n+1 are strongly related to algebraic equations of degee n+1. Using this relations, we give explicit monodromy representations of the differential equations.

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In this paper we study coupled systems of semilinear wave equations and derive sharp conditions for the small data global existence and blowup for the system. The way of the interaction in the nonlinearities plays an important role to determine the condition. We focus on the case where propagation speeds also come into play. The discrepancy of the speeds is actually essential in Theorem 3.1 for instance. Moreover, in some cases we have different conclusion for the same nonlinearity according to the order of them. To handle such cases, we modify the argument presented by F. John [12].

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We prove that the semilinear Dirichlet problem for a Laplace equation on a unit ball, involving the nonlinearity f(r,u)=-a(r)u+b(r)u^p, with a subcritical p, has a unique positive solution, provided a(r) is positive, increasing and convex, while b(r) is positive, decreasing and concave. Moreover, we prove that this solution is non-degenerate. We also present a uniqueness result in case a(r) is negative.

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This paper is a continuation of our early paper [MS]. In this paper, we study the structure of formal power series solutions of a first order nonlinear partial differential equation which is defined and holomorphic in a neighborhood of the origin of several complex variables. Our main theorem (Theorem 1.1) characterizes the convergence or the divergence of a given formal power series solution a priori. Especially, in the case of divergence, we give the rate of divergence in terms of Gevrey index which is known as Maillet type theorem (Theorem 1.1, (ii)). It should be mentioned that the notion of singularity or singular point for nonlinear equations depends on each solution, and the coexistence of a convergent solution and a divergent solution for a nonlinear equation is possible.

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We investigate the solutions of the fourth Painlevé equation in two variables by means of tau functions. Hirota bilinear forms and birational symmetries of the equation are presented. We obtain in particular a complete description of the rational solutions in terms of Schur polynomials.

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We present a simple heuristic method to derive the Painlevé differential equations from the corresponding geometry of rational surfaces.

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