In this article, we study the narrow capture problem on a Riemannian 2-manifold. This involves the derivation of the mean first passage (sojourn) time of a surface-bound ion modeled as a Brownian particle. We use a layer potential argument in conjunction with microlocal analysis in order to derive the leading order singularity as well as the $O(1)$ term of the mean first passage time and the associated spatial average.

We consider the Cauchy problem derived from an attraction-repulsion chemotaxis system in $\mathbb R^n$. This system consists of the partial differential equations of the parabolic-elliptic-elliptic type, and possesses two components representing the chemoattractant and chemorepellent. In this paper, we show that the system for the three or four-dimensional case admits the solution which blows up in finite time in the attractive dominant case by use of the moment method.

The present paper is concerned with the Sturm-Liouville eigenvalue problems. The accurate asymptotic behavior of the first eigenvalues on the jump set is given. The results indicate that the asymptotic behaviors are only related to the values of the coefficients of the equations in the neighborhood of the end points. With an application on the nonlinear propagation phenomena, we discuss the monotonicity of the first eigenvalues with respect to the parameter involved in both equations and boundary conditions. The infimum of the parameter which guarantees the monotonicity of the first eigenvalues is obtained.

In the present work, we are concerned with the existence of normalized solutions to the following Schrödinger-Poisson System $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u + \mu (\ln|\cdot|\ast |u|^{2})u = f(u) \ \text{ in } \mathbb{R}^2 , \\ \int_{\mathbb R^2} |u(x)|^2 dx = c,\ c > 0 , \end{array} \right. $$ for $\mu \in \mathbb{R} $ and a nonlinearity $f$ with exponential critical growth. Here, $\lambda\in \mathbb{R}$ stands as a Lagrange multiplier and it is part of the unknown. Our main results extend and/or complement some results found in [3] and [13].

We prove that the integral operators associated with the layer heat potentials depend smoothly upon a parametrization of the support of integration. The analysis is carried out in the optimal H\"older setting.

The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$ \partial_t u-\Delta u^m+|x|^{\sigma}u^q=0, \qquad (t,x)\in(0,\infty)\times \mathbb R ^N, $$ with $m\geq1$, $q\in(0,1)$ and $\sigma > 0$, is addressed. Introducing the critical exponent $\sigma^* := 2(1-q)/(m-1)$ for $m > 1$ and $\sigma^*=\infty$ for $m=1$, extinction in finite time is known to take place for $\sigma\in [0,\sigma^*)$ and an alternative proof is provided therein. When $m > 1$ and $\sigma\ge \sigma^*$, the occurrence of finite time extinction is proved for a specific class of initial conditions, thereby supplementing results on non-extinction that are available in that range of $\sigma$ and showing their sharpness.