We consider large time behavior of solutions to the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We treat the case in which the nonlinearity contains non-oscillating factor $|u|^{1+2/d}$. The case is excluded in our previous studies. It turns out that there are no solutions that behave like a free solution with or without logarithmic phase corrections. We also prove nonexistence of an asymptotic free solution in the case that the gauge invariant nonlinearity is dominant, and give a finite time blow-up result.

This paper examines the existence and nonexistence of solutions on the Cauchy problem for the evolutionary p-Laplacian equation with nonlocal source. By a priori estimates, we establish the local existence, global existence and nonexistence of solutions for that Cauchy problem. In particular, a Fujita's type critical exponent is obtained, which extends several classical results to the problem considered here.

This article addresses the large data global regularity for the equivariant case of the $2+1$-dimensional Faddeev model and shows that it holds true for initial data in $H^s\times H^{s-1}(\mathbb R^2)$ with $s>3$.

We consider some removability problem for solutions to the so-called $k$-Hessian equation and $k$-curvature equation. We prove that if a $C^1$ function $u$ is a generalized solution to $k$-Hessian equation $F_k[u]=g(x,u,Du)$ or $k$-curvature equation $H_k[u]=g(x,u,Du)$ in $\Omega \setminus u^{-1}(E)$ for $E \subset \mathbb{R}$, then it is indeed a generalized solution to the same equation in the whole domain $\Omega$, under some hypotheses on $u$, $g$ and $E$.

In this paper, we prove an existence result for weak solutions to a class of Dirichlet boundary value problems whose prototype is \begin{equation*} \label{pa} \left\{ \begin{array}{lll} -\Delta_p u =\beta |\nabla u|^{q} +c(x)|u|^{p-2}u +f & & \text{in}\ \Omega \\ u=0 & & \text{on}\ \partial \Omega , \end{array} \right. \end{equation*} where $\Omega $ is a bounded open subset of $\mathbb R^N$, $N\geq 2$, $\Delta_p u={\rm div} \left(|\nabla u|^{p-2}\nabla u\right)$, $1 < p < N$, $ p-1 < q\le p-1+\frac p N$, $\beta $ is a positive constant, $c\in L^{\frac N p}(\Omega)$ with $c\ge 0$, $c\neq 0$ and $f\in L^{(p^*)'}(\Omega).$ We further assume smallness assumptions on $c$ and $f$. Our approach is based on Schauder's fixed point theorem.

We study the global existence of solutions to semilinear damped wave equations in the scattering case with power-type nonlinearity on the derivatives, posed on nontrapping asymptotically Euclidean manifolds. The main idea is to shift initial time by local existence. As a result, we could convert the damping term to small enough perturbation and obtain the global existence.