Related papers: Singularities of complex-valued solutions to linea…
We construct examples of finite time singularity from smooth data for linear uniformly parabolic systems in the plane. We obtain similar examples for quasilinear systems with coefficients that depend only on the solution.
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 short paper, we are concerned with the blowup phenomenon of stochastic parabolic equations. By using comparison principle and the results of deterministic parabolic equations, we obtain blowup results of solutions for stochastic…
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…
We give blow-up analysis for the solutions of an elliptic equation under some conditions. Also, we derive a compactness result for this equation.
This paper provides the upper and lower bounds of blowup time and blowup rate as well as the exponential growth estimate of blowup solutions for a pseudo-parabolic equation with singular potential. These results complement the ones obtained…
We consider the blow-up sets and the upper blow-up rate estimates for two parabolic problems defined in a ball.
We consider the classical problem of the blowing-up of solutions of the nonlinear heat equation. We show that there exist infinitely many profiles around the blow-up point, and for each integer $k$, we construct a set of codimension $2k$ in…
An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime…
We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + \lambda c(u)c'(u)( u_x)^2$ with the real parameter $\lambda$. In previous works, it was reported that there exist…
We establish the asymptotics of blowup for nonlinear heat equations with superlinear power nonlinearities in arbitrary dimensions and we estimate the remainders.
Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result…
This paper deals with blow-up for the complex-valued semilinear wave equation with power nonlinearity in dimension 1. Up to a rotation of the solution in the complex plane, we show that near a characteristic blow-up point, the solution…
We study a dissipative nonlinear equation modelling certain features of the Navier-Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions $n\leq 4$.…
In this paper we prove that for a certain class of initial data, smooth solutions of the hydrostatic Euler equations blow up in finite time.
We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any…
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 a blow-up solution for the complex-valued semilinear wave equation with power nonlinearity in one space dimension. We first characterize all the solutions of the associated stationary problem as a two-parameter…
In this work, we study the numerical solution for parabolic equations whose solutions have a common property of blowing up in finite time and the equations are invariant under the following scaling transformation $$u \mapsto…
We are concerned with the qualitative analysis of positive singular solutions with blow-up boundary for a class of logistic-type equations with slow diffusion and variable potential. We establish the exact blow-up rate of solutions near the…