We construct finite positive time blow up solutions for the nonlinear Schrödinger equation with the power nonlinearity whose coefficient is complex number. We also observe that those solutions exist time globally for the negative time. We show a sequence of solutions closes to the blow up profile which is a blow up solution of ODE. We apply the Aubin-Lions lemma for the compactness argument for its convergence.

We build a discrete stochastic process adapted to the (nonlinear) dominative $p$-Laplacian $$ \mathcal{D}_p u(x):=\Delta u + (p-2)\lambda_{N} , $$ where $\lambda_{N}$ is the largest eigenvalue of $D^2 u$ and $p > 2$. We show that the discrete solutions of the Dirichlet problems at scale $\varepsilon$ tend to the solution of the Dirichlet problem for $\mathcal{D}_p$ as $\varepsilon\to 0$. We assume that the domain and the boundary values are both Lipschitz.

We consider the orbital stability of line solitons of the Kadomtsev–Petviashvili-I equation in $\mathbb R \times (\mathbb R/2\pi\mathbb Z)$. Zakharov [40] and Rousset–Tzvetkov [31] proved the orbital instability of the line solitons of the Kadomtsev–Petviashvili-I equation on $\mathbb R^2$. The orbital instability of the line solitons on $\mathbb R \times (\mathbb R/2\pi\mathbb Z)$ with the traveling speed $c > {\frac {4}{\sqrt{3}}} $ was proved by Rousset–Tzvetkov [32] and the orbital stability of the line solitons with the traveling speed $0 < c < {\frac {4}{\sqrt{3}}} $ was showed in [34]. In this paper, we prove the orbital stability of the line soliton of the Kadomtsev–Petviashvili-I equation on $\mathbb R \times (\mathbb R/2\pi\mathbb Z)$ with the critical speed $c= {\frac {4}{\sqrt{3}}} $ and the Zaitsev solitons near the line soliton. Since the linearized operator around the line soliton with the traveling speed $ {\frac {4}{\sqrt{3}}} $ is degenerate, we cannot apply the argument in [32, 33, 34]. To prove the stability, we investigate the branch of the Zaitsev solitons and apply the argument [37].

In the present paper, we first study the nonexistence of positive solutions of the following nonlinear parabolic problem \begin{equation*} \begin{cases} \frac{\partial u}{\partial t}=\Delta_g( u^m)+V(x)u^m+\lambda u^q & \text{in}\quad \Omega \times (0, T ), \\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T). \end{cases} \end{equation*} Here, $\Omega$ is a bounded domain with smooth boundary in a complete non-compact Riemannian manifold $M$, $0 < m < 1$, $V\in L_{\text{loc}}^1(\Omega)$, $ q > 0 $ and $\lambda\in \mathbb{R}$. Next, we prove some Hardy and Leray type inequalities with remainders on a Riemannian Manifold $M$. Furthermore, we obtain explicit (sometimes optimal) constants for these inequalities and present several nonexistence results with help of Hardy and Leray type inequalities on the hyperbolic space $\mathbb{H}^n$.

The objective of the paper is to characterize the Gaussian free field as a stationary solution of the heat equation with additive space-time white noise. In the whole space, the investigation leads to other types of Gaussian fields, as well as interesting phenomena in dimensions one and two.