Related papers: Singular Phenomena of Solutions for Nonlinear Diff…
We study the homogeneous Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $\partial_t u =-\mathcal{L} u^m$ posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with…
In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping…
In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system \begin{equation*} \left\{ \begin{array}{l} \begin{aligned} &u_t = \Delta u - \nabla(u f(|\nabla v|^2 )\nabla v), \\[6pt] &0= \Delta v…
In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic $p(.)$-Laplacian type equation with logarithmic nonlinearity: $$ u_t-\Delta u_t-\mbox{div}\left(\left\vert \nabla…
This paper concerns the initial-boundary value problem for a mixed pseudo-parabolic $p$-Laplacian type equation. By constructing a family of potential wells, we first present the explicit expression for the depth of potential well, and then…
We investigate gradient flows of some homogeneous functionals in R^N , arising in the Lagrangian approximation of systems of self-interacting and diffusing particles. We focus on the case of negative homogeneity. In the case of strong…
We consider the initial boundary value problem in exterior domain for strongly damped wave equations with power type nonlinearity |u|^p. We will establish blow-up results under some conditions on the initial data and the exponent p.
We study systems of nonlinear ordinary differential equations where the dominant term, with respect to large spatial variables, causes blow-ups and is positively homogeneous of a degree $1+\alpha$ for some $\alpha>0$. We prove that the…
Let $G=(V,E)$ be a locally finite connected weighted graph, $\Delta$ be the usual graph Laplacian. In this paper, we study the blow-up problems for the nonlinear parabolic equation $u_t=\Delta u + f(u)$ on $G$. The blow-up phenomenons of…
We study the blow-up question for the diffusion equation involving a nonlocal derivative in time defined by convolution with a nonnegative and nonincreasing kernel, and a nonlocal operator in space driven by a nonnegative radial L\'evy…
Motivated by a nonlocal free boundary problem, we study uniform properties of solutions to a singular perturbation problem for a boundary-reaction-diffusion equation, where the reaction term is of combustion type. This boundary problem is…
In this article, we investigate the initial and boundary blow-up problem for the $p$-Laplacian parabolic equation $u_t-\Delta_p u=-b(x,t)f(u)$ over a smooth bounded domain $\Omega$ of $\mathbb{R}^N$ with $N\ge2$, where $\Delta_pu={\rm…
In this paper, the existence, uniqueness and dependence on initial value of solution for a singular diffusion equation with nonlinear boundary condition are discussed. It is proved that there exists a unique global smooth solution which…
This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $\Omega$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$,…
The fourth-order quasilinear reaction-diffusion equation with a p-Laplacian operator is shown to admit three types of blow-up. Self-similar patterns are first constructed for the regional blow-up case, where the rescaled problem admits a…
Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on the dynamics of the singularities in the complexified…
In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that…
In this paper we consider the nonlinear dispersive wave equation on the real line, $u_t-u_{txx}+[f(u)]_x-[f(u)]_{xxx}+\bigl[g(u)+\frac{f''(u)}{2}u_x^2\bigr]_x=0$, that for appropriate choices of the functions $f$ and $g$ includes well known…
In this paper we report on numerical studies of formation of singularities for the semilinear wave equations with a focusing power nonlinearity $u_{tt} - \Delta u = u^{p}$ in three space dimensions. We show that for generic large initial…
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…