Finite-time blow-up of solutions to a cancer invasion mathematical model with haptotaxis effects

This paper considers the nonlinear system of cancer invasion model with haptotaxis effects in a bounded domain $\Omega ?{\mathbb{R}}^n$ under the homogeneous Neumann and Robin type boundary conditions. This model also includes the growth and decay effects of cancer cells, normal cells and matrix degrading enzymes. This study devoted to estimate the lower bounds for the finite time blow up of non-negative solutions of the given cancer invasion system using first order differential inequality techniques and certain inequalities.     

一类半线性热方程整体解的非存在性

用高斯函数方法证明了一类半线性热方程初值问题整体解的非存在性与有限时间Ｂｌｏｗ－ｕｐ。     

Dynamical low-rank approximation: applications and numerical experiments

Dynamical low-rank approximation is a differential-equation-based approach to efficiently compute low-rank approximations to time-dependent large data matrices or to solutions of large matrix differential equations. We illustrate its use in the following application areas: as an updating procedure in latent semantic indexing for information retrieval, in the compression of series of images, and in the solution of time-dependent partial differential equations, specifically on a blow-up problem of a reaction-diffusion equation in two and three spatial dimensions. In 3D and higher dimensions, space discretization yields a tensor differential equation whose solution is approximated by low-rank tensors, effectively solving a system of discretized partial differential equations in one spatial dimension.     

Blow-up time estimates and simultaneous blow-up of solutions in nonlinear diffusion problems

This paper deals with some homogeneous Dirichlet problems of nonlinear diffusion equations. After demonstrating the existence and uniqueness of weak solutions, we prove the existence and non-existence of global solutions. The influence of coefficients and geometry of domain is shown clearly on the existence of global solutions. We give the complete classification of simultaneous blow-up of solutions with blow-up rates as well in one-dimensional space. Moreover, the bounds of blow-up time are studied for all dimensions of space domain.     

Nonlinear Hyperbolic Rigid Heat Conductor of the Coleman Type

A one-dimensional nonlinear hyperbolic homogeneous isotropic rigid heat conductor proposed by Coleman is analyzed using the method of weakly nonlinear geometric optics. For such a model the law of conservation of energy, the dissipation inequality, the Cattaneo's equation, and a generalized energy-entropy relation with a parabolic variation of the energy and entropy along the heat-flux axis, are postulated. First, it is shown that the model can be described by a non-homogeneous quasi-linear hyperbolic matrix partial differential equation of the first order for an unknown vector u = (θ, Q) ^T , where θ and Q are the dimensionless absolute temperature and heat-flux fields, respectively. Next, the Cauchy problem for the matrix equation with a weakly perturbed initial condition is formulated, and an asymptotic solution to the problem in terms of the amplitudes σ_α (α = 1, 2) that satisfy a pair of nonlinear first order partial differential equations, is obtained. The Cauchy problem is then solved in a closed form when the initial data are suitably restricted. Numerical examples are included.     

广义形式Carleman方程组的Cauchy问题

本文证明了广义形式的Ｃａｒｌｅｍａｎ方程组Ｃａｕｃｈｙ问题整体光滑解的整体存在性定理以及解的破裂现象。     

Blow-up analysis in quasilinear reaction–diffusion problems with weighted nonlocal source

In this paper, we consider the blow-up of solutions to a class of quasilinear reaction–diffusion problems

where $\Omega$ is a bounded convex domain in ${\mathbb{R}}^n(n\ge 2)$, weighted nonlocal source satisfies $a\left(x\right)f\left(u(x,t)\right)\le a_1+a_2{\left(u(x,t)\right)}^p{\left({\int }_{\Omega }{\left(u(x,t)\right)}^l\mathrm{d}x\right)}^m,$ and $a_1,a_2,p,l,$ and $m$ are positive constants. By utilizing a differential inequality technique and maximum principles, we establish conditions to guarantee that the solution remains global or blows up in a finite time. Moreover, an upper and a lower bound for blow-up time are derived. Furthermore, two examples are given to illustrate the applications of obtained results.     

Stochastic quasilinear viscoelastic wave equation with degenerate damping and source terms

In this paper, we consider a stochastic quasilinear viscoelastic wave equation with degenerate damping and source term. We prove the blow-up of solution for stochastic quasilinear viscoelastic wave equation with positive probability or explosive in energy sense.     

On the blow-up criterion for the quasi-geostrophic equations in homogeneous Besov spaces

In this paper, we consider the blow-up criterion for the quasi-geostrophic equations with dissipation ${\Lambda }^{\gamma }$ ($0<\gamma <1$). By establishing a new trilinear estimate, we show that if

$\theta \in L^{\frac{\gamma }{\gamma +s?1}}(0,T;{\stackrel{?}{B}}_{\infty ,\infty }^s\left({\mathbb{R}}^2\right))$

for some $s\in \left(1?\frac{\gamma }{2},1\right)$, then the solution can be extended smoothly past $T$. This improves and extends the corresponding results in Dong and Pavlovi? (2009) ([32]) and Yuan (2010).     

聚四氟乙烯热收缩管的加工技术

介绍了聚四氟乙烯材料经推压、烧结、热处理、吹胀等工艺成型热收缩管的加工技术