Related papers: Blowup of Solutions of the Hydrostatic Euler Equat…
Presented are two results on the formation of finite time singularities of solutions to the compressible Euler equations in two and three space dimensions for isentropic, polytropic, ideal fluid flows. The initial velocity is assumed to be…
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 $p>1$,…
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…
We consider the nonlinear heat equation with a nonlinear gradient term: $\partial_t u =\Delta u+\mu|\nabla u|^q+|u|^{p-1}u,\; \mu>0,\; q=2p/(p+1),\; p>3,\; t\in (0,T),\; x\in \R^N.$ We construct a solution which blows up in finite time…
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…
In the note, a certain scenario of potential Type II blowups of axisymmetric solutions to the Navier-Stokes equations is considered. The main tool of the treatment of such blowups is the corresponding Euler scaling.
We give sufficient conditions on the initial data so that a semilinear wave inequality blows-up in finite time. Our method is based on the study of an associated second order differential inequality. The same method is applied to some…
In this paper, we consider the Cauchy problem of the incompressible liquid crystal equations in $n$ dimensions. We prove the local well-posedness of mild solutions to the liquid crystal equations with $L^\infty$ initial data, in particular,…
In this paper, we consider the complex Ginzburg-Landau equation $$ \partial_t u = (1 + i \beta) \Delta u + (1 + i \delta) |u|^{p-1}u - \alpha u, \quad \text{where } \beta, \delta, \alpha \in \mathbb{R}. $$ The study focuses on investigating…
This paper is devoted to the analysis of blow-up solutions for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities \[ iu_{t}+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u. \] When $p_1=\frac{4}{N}$ and…
The paper studies the possible blowup of the total variation for entropy weak solutions of the p-system, modeling isentropic gas dynamics. It is assumed that the density remains uniformly positive, while the initial data can have…
In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective…
We consider the six dimensional energy-critical semilinear heat equation with self-similarly decaying initial data. Our main result shows the existence of sign-changing solutions that exhibit infinite-time blow-up and nonnegative solutions…
We consider Einstein's equations coupled to the Euler equations in plane symmetry, with compact spatial slices and constant mean curvature time. We show that for a wide variety of equations of state and a large class of initial data,…
Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat de Sitter spacetime. We show that blow-up in a finite time occurs for the equation with arbitrary power nonlinearity as well as upper…
In this paper we prove that all initially-smooth solutions of the Euler-Weil-Petersson equation, which describes geodesics on the universal Teichm\"uller space under the Weil-Petersson metric, will remain smooth for all time. This extends…
We consider the porous medium equation with power-type reaction terms $u^p$ on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If $p>m$, small data give rise to…
In this paper, we consider the finite time blow up of solutions for the following two kinds of nonlinear wave equations on de Sitter spacetime \begin{eqnarray*} &&\square_g=F(u),\\ &&\square_g=F(\partial_tu,\nabla u). \end{eqnarray*} This…
We prove finite-time blowup of classical solutions for the compressible Upper Convective Maxwell (UCM) viscoelastic fluid system. By establishing a key energy identity and adapting Sideris' method for compressible flows, we derive a…
This paper deals with the blow-up properties of the solutions of the semilinear heat equation