We consider a nonlinear Neumann eigenvalue problem driven by the p-Laplacian and with a (p – 1)-sublinear reaction. Using variational methods together with suitable truncation techniques, we prove a bifurcation-type theorem for the eigenvalue problem. Namely, we show that there is a critical parameter value λ_* > 0 such that for all λ > λ_* the problem has at least two positive solutions, for λ = λ_* there is at least one positive solution and for λ ∈ (0, λ_*) no positive solutions exist.

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We consider the Cauchy problem of the fifth order KdV equation with low regularity data. We cannot apply the iteration argument to this problem when initial data is given in the Sobolev space H^s for any s ∈ R. So we give initial data in H^s,a = H^s ∩ H^a with a ≤ min{s, 0}. Then we recover more derivatives of the nonlinear term to be able to use the iteration method. Therefore we obtain the local well-posedness in H^s,a in the case s ≥ max{–1/4, –2a – 2}, –3/2 < a ≤ a –1/4 and (s,a) ≠ (–1/4, –7/8). Moreover, we obtain the ill-posedness in some sense when s < max{–1/4, –2a – 2}, a ≤ –3/2 or a > –1/4. The main tool is a variant of the Fourier restriction norm method, which is based on Kishimoto's work (2009).

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Let Ω be a bounded domain in R^N (N > 2). We are concerned with the existence and nonexistence of solutions for the following nonlocal problem, –M (∫_Ω|∇u(x)|²dx)Δu = |u|^p–1u + λf(x) in Ω, u_|∂Ω = 0. Where M is continuous function on R⁺ and f ∈ C¹($\overline{\Omega}$) changes sign. λ and p are positive parameters. By direct variational method, Galerkin approach and sub and super solutions method some results are established.

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Chazy studied a family of homogeneous third-order autonomous differential equations. They are those, within a certain class, admitting exclusively single-valued solutions. Each one of these equations yields a polynomial vector field in complex three-dimensional space. For almost all of these vector fields, the Zariski closure of a generic orbit yields an affine surface endowed with a holomorphic vector field that has exclusively single-valued solutions. We classify these surfaces and relate this classification to recent results of Rebelo and the author.

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An ODE in conditions of Peano's theorem has been considered. A family of solutions with Borel measurable dependence of the initial data has been obtained. Corresponding existence theorem for the transport equation has been proved.

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We investigate the Cauchy problem for second-order hyperbolic operators in the framework of the space of C^∞ functions. In the case where the coefficients of their principal parts depend only on the time variable and are real analytic, we give a sufficient condition for C^∞ well-posedness, which is also a necessary one when the space dimension is less than 3 or the coefficients of the principal parts are semi-algebraic functions (e.g., polynomials) of the time variable.

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We consider a curve flow for maps from a real line into a compact almost Hermitian manifold, which is governed by a third order nonlinear dispersive equation. This article shows short-time existence of a solution to the initial value problem for the equation. The difficulty comes from the lack of the Kähler condition on the target manifold, since the covariant derivative of the almost complex structure causes a loss of one derivative in our equation and thus the classical energy method breaks down in general. In the present article, we can overcome the difficulty by constructing a gauge transformation on the pull-back bundle for the map to eliminate the derivative loss essentially, which is based on the local smoothing effect of third order dispersive equations on the real line.

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We consider the Cauchy problem for the Kadomtsev-Petviashvili equations u_t + u_xxx + σ∂_x^–1u_yy = –(u²)_x, (x, y) ∈ R², t ∈ R, u(0, x, y) = u₀(x, y), (x, y) ∈ R², where σ = 1 or σ = –1, ∂_x^–1 = ∫_–∞^x dx′. We prove that the maximal existence time T is estimated from below as T ≥ exp(C/ε), where ε denotes the size of the initial data, C > 0 is a constant.

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