This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on R^d. If d = 2, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space H^s(R²) for s ≥ 1/4. If d ≥ 3, by employing U^p and V^p spaces, we establish the small data global well-posedness in the scaling critical Sobolev space H^s_c(R^d) where s_c = d/2-1.

We introduce two variants of the q-hypergeometric equation. We obtain several explicit solutions of the variants of the q-hypergeometric equation. We show that a variant of the q-hypergeometric equation can be obtained by a restriction of the q-Appell equation of two variables.

We consider the Cauchy problem for nonlinear Schrödinger equations with fourth-order anisotropic dispersion. In this paper, we show the existence of blow-up solutions in the mass-supercritical case and in the mass-critical case.

In this paper, we study existence of positive and symmetric solutions for some boundary value problem on a spherical cap of unit ball in four-dimensional space. It seems that this problem has been studied for the equation containing critical Sobolev exponent for three-dimensional space; so that we consider non-critical case in this paper. Then for subcritical case, we can show the existence and non-existence result for such solution completely if the spherical cap is contained in the hemisphere. In addition, we get some partial result if the spherical cap contains the hemisphere.