In this paper, we consider the Cauchy problem for the nonlinear higher order Schrödinger equations on modulation spaces $M_{p,q}^s$ and show the existence of a unique global solution by using integrability of time decay factors of time decay estimates. As a result, we are able to deal with wider classes of a nonlinearity and a solution space. Moreover, we study time decay estimates of a semi--group $e^{it\phi(\sqrt{-\Delta})}$ with a polynomial symbol $\phi$. Considering multiplicities of critical points and inflection points of $\phi$ carefully, we have time decay estimates with better time decay rate.

The semigroup of a generalized initial value problem including, as a particular case, the Cox-Ingersoll-Ross (CIR) equation for the price of a zero-coupon bond, is studied on spaces of continuous functions on $[0,\infty]$. The main result is the first proof of the strong continuity of the CIR semigroup. We also derive a semi-explicit representation of the semigroup and a Feynman-Kac type formula, in a generalized sense, for the unique solution of the CIR initial value problem as a useful tool for understanding additional properties of the solution itself. The Feynman-Kac type formula is the second main result of this paper.

This paper is concerned with the stability of a parallel flow of the compressible Navier-Stokes equation in a cylindrical domain. The spectrum of the linearized operator is analyzed for the purpose of the study of the nonlinear stability. It is shown that, if the Reynolds and Mach numbers are sufficiently small, then the linearized semigroup is decomposed into two parts; one behaves like a solution of a one dimensional heat equation as time goes to infinity and the other one decays exponentially. Some estimates related to the spectral projections are established, which will also be useful for the study of the nonlinear problem.

Nonlinear, dispersive wave equations arise as models of various physical phenomena. A major preoccupation on the mathematical side of the study of such equations has been to settle the fundamental issues of local and global well-posedness in Hadamard's classical sense. The development so far has been mostly for the initial-value problem for single equations.

However, systems of such equations have also received consideration, and there is now theory for pure initial-value problems where data are given on the entire space or on the torus. Here, consideration is given to non-homogeneous initial-boundary-value problems for a class of BBM-type systems having the form \begin{equation*} \begin{aligned} u_t + u_x -u_{xxt} + P(u,v)_x \, = \, 0, \\ v_t + v_x - v_{xxt} + Q(u,v)_x \, = \, 0, \end{aligned} \end{equation*} where $P$ and $Q$ are homogeneous, quadratic polynomials, $u$ and $v$ are real-valued functions of a spatial variable $x$ and a temporal variable $t$, and subscripts connote partial differentiation. Local in time well-posedness is established in the quarter plane $\{(x,t): x \geq 0, \, t \geq 0 \}$. Under certain restrictions on the coefficients of the nonlinearities $P$ and $Q$, global well posedness is also shown to obtain.

We prove scattering of solutions below the energy norm of the nonlinear Klein-Gordon equation in 3D with a defocusing power-type nonlinearity that is superconformal and energy subcritical: this result extends those obtained in the energy class [4, 18, 19] and those obtained below the energy norm under the additional assumption of spherical symmetry [25]. In order to do that, we generate an exponential-type decay estimate in $H^{s}$, $s < 1$, by means of concentration [1] and a low-high frequency decomposition [2, 7]: this is the starting point to prove scattering. On low frequencies, we modify the arguments in [18, 19]; on high frequencies, we use the smoothing effect of the solutions to control the error terms: this, combined with an almost conservation law, allows to prove this decay estimate.

We consider the pseudo-$p$-Laplacian operator: $$ \tilde \Delta_p u = \sum_{i=1}^N \partial_i (|\partial_i u|^{p-2} \partial_i u)=(p-1) \sum_{i=1}^N |\partial_i u|^{p-2} \partial_{ii} u \ \text{ for $p > 2$.} $$ We prove interior regularity results for the viscosity (resp. weak) solutions in the unit ball $B_1$ of $\tilde \Delta_p u =(p-1) f$ for $f\in { \mathcal C} (\overline{B_1})$ (resp. $f\in L^\infty(B_1)$). First, the Hölder local regularity for any exponent $\gamma < 1$, recovering in that way a known result about weak solutions. Second, we prove the Lipschitz local regularity.