Related papers: The $C^0$-convergence at the Neumann boundary for …
In a recent series of important works \cite{wei-zhang-1,wei-zhang-2,wei-zhang-3}, Wei-Zhang proved several vanishing theorems for non-simple blow-up solutions of singular Liouville equations. It is well known that a non-simple blow-up…
In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power non-linearity in one space dimension. We show that the set of non characteristic points $I_0$ is open and that the blow-up curve is of…
We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-\Delta u= \lambda V(x) e^u-4\pi N \delta_0\;\mbox{ in } B_1,\quad u=0 \;\mbox{ on }\partial B_1,$$ where $B_1$ is the unit ball in…
For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions as far as blowup points are either regular points or non-quantized singular sources. In particular the uniqueness result…
In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis,…
We consider a non-local Liouville equation corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up at a critical point of the harmonic extension of the prescribed curvature…
The study of blow-up solution of time-fractional heat equations is of significant and wide-ranging interest for its multitude of applications. These types of equations are used to model several real problems in science and engineering. This…
In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension $1$. More precisely, given a sequence $u_k :\mathbb{R} \to \mathbb{R}$ of solutions to \begin{equation} (-\Delta)^\frac{1}{2} u_k…
We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by…
We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-\Delta u= \la a(x) e^u-4\pi N \delta_0\;\hbox{ in } \Omega,\quad u=0 \;\hbox{ on }\partial \Omega,$$ where $\Omega$ is a smooth and…
We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of…
In this article we introduce a new blowup criterion for (generalized) Euler-Arnold equations on $\mathbb R^n$. Our method is based on treating the equation in Lagrangian coordinates, where it is an ODE on the diffeomorphism group, and…
We consider the nonlocal Liouville type equation $$ (-\Delta)^{\frac{1}{2}} u = \varepsilon \kappa(x) e^u, \quad u > 0, \quad \mbox{in } I, \qquad u = 0, \quad \mbox{in } \mathbb{R} \setminus I, $$ where $I$ is a union of $d \geq 2$…
In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine…
We investigate the blow-up behavior and Liouville-type theorems of solutions to a class of generalized Camassa-Holm-Kadomtsev-Petviashvili (CH-KP) equations with a generally smooth nonlinear term $g(u)$. First, using the continuation…
This paper is motivated by the question of whether a sequence of solutions of a given integrable system can be blown up to obtain a solution of a different integrable system in the limit. We study a specific example of this phenomenon.…
For Liouville equation with quantized singular sources, the non-simple blowup phenomenon has been a major difficulty for years. It was conjectured by the first two authors that the non-simple blowup phenomenon does not occur if the equation…
On a compact Riemann surface $(\Sigma, g)$ with a smooth boundary $\partial \Sigma$, we consider the following mean field equations with Neumann boundary conditions: $$ -\Delta_g u = \lambda \left(\frac{Ve^u}{\int_{\Sigma} Ve^u \, dv_g} -…
In this paper we study a simple non-local semilinear parabolic equation with Neumann boundary condition. We give local existence result and prove global existence for small initial data. A natural non increasing in time energy is associated…
This work studies nonnegative solutions for the Cauchy, Neumann, and Dirichlet problems of a logistic type reaction-diffusion equation. The finite time blowup results for nonnegative solutions under various restrictions on the coefficients…