In this paper, we consider the Cauchy problem of a degenerate parabolic equation not in divergence form with weighted source in Rn, and the weighted source is positive and continuous in Rn. By extending the classical result of Fujita, we find the second critical exponent for the existence of global and nonglobal solutions with n ≥ 1, including the results of Yang, Zheng, and Zhou, which give the second critical exponent with n ≥ 2.

In this note we establish existence and uniqueness of weak solutions of the linear elliptic equation div[A(x)∇u] = div F(x), where the matrix A is just measurable and its skew-symmetric part can be unbounded. Global reverse Hölder's regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing examples, that boundedness and ellipticity of A are not sufficient for higher integrability estimates even when the symmetric part of A is the identity matrix. In addition, the examples also show the necessity of the dependence of α in the Hölder Cα-regularity theory on the BMO semi-norm of the skew-symmetric part of A. The paper is an extension of classical results obtained by N. G. Meyers (1963) in which the skew-symmetric part of A is assumed to be zero.

We consider a parametric Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which has singular and critical terms. Using cut-off techniques and variational methods, we show that for all small values of the parameter λ > 0, the problem has a positive solution and this solution converges to 0 in as λ → 0+.

In this paper we prove a priori Hölder and Lipschitz regularity estimates for viscosity solutions equations governed by the inhomogeneous infinite Laplace operator relative to a frame of vector fields.

We prove low regularity a priori estimates for the derivative nonlinear Schr\"{o}dinger equation in Besov spaces with positive regularity index conditional upon small $L^2$-norm. This covers the full subcritical range. We use the power series expansion of the perturbation determinant introduced by Killip--Vi\c{s}an--Zhang for completely integrable PDE. This makes it possible to derive low regularity conservation laws from the perturbation determinant.

In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by a given polynomial. We apply this spectral theory to study the asymptotic behavior of solutions of fractional linear difference equations. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that the α-resolvent operator that is associated with the fractional equation, satisfies an asymptotic estimate of Katznelson-Tzafriri type, provided that it is bounded by the polynomial, and the spectrum of the fractional equation on the unit circle is either empty or consists of only one element 1. Three concrete examples are also included to illustrate the obtained results.

We consider the energy-critical stochastic cubic nonlinear Schrödinger equation on R4 with additive noise, and with the non-vanishing boundary conditions at spatial infinity. By viewing this equation as a perturbation to the energy-critical cubic nonlinear Schrödinger equation on R4, we prove global well-posedness in the energy space. Moreover, we establish unconditional uniqueness of solutions in the energy space.

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

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.