Finite time blow-up for a class of parabolic or pseudo-parabolic equations

In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations:

$u_t?a\Delta u_t?\Delta u+bu=k\left(t\right)|u{|}^{p?2}u,(x,t)\in \Omega \times (0,T),$

where $a\ge 0$, $b>?{?}_1$ with ${?}_1$ being the principal eigenvalue for $?\Delta$ on $H_0^1\left(\Omega \right)$ and $k\left(t\right)>0$. By using the potential well method, Levine’s concavity method and some differential inequality techniques, we obtain the finite time blow-up results provided that the initial energy satisfies three conditions: (i) $J(u_0;0)<0$; (ii) $J(u_0;0)\le d\left(\infty \right)$, where $d\left(\infty \right)$ is a nonnegative constant; (iii) $0<J(u_0;0)\le C\rho \left(0\right)$, where $\rho \left(0\right)$ involves the $L^2$-norm or $H_0^1$-norm of the initial data. We also establish the lower and upper bounds for the blow-up time. In particular, we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level.     

Nonexistence of global solutions for a class of nonlocal in time and space nonlinear evolution equations

In this paper, we study the nonlocal nonlinear evolution equation

${}^C{D}_{0|t}^{\alpha }u(t,x)?(J1\left|u\right|?\left|u\right|)(t,x){+}^C{D}_{0|t}^{\beta }u(t,x)={\left|u(t,x)\right|}^p,t>0,x\in {\mathbb{R}}^d,$

where $1<\alpha <2$, $0<\beta <1$, $p>1$, $J:{\mathbb{R}}^d\to {\mathbb{R}}_{+}$, $1$ is the convolution product in ${\mathbb{R}}^d$, and ${}^C{D}_{0|t}^q$, $q\in \{\alpha ,\beta \}$, is the Caputo left-sided fractional derivative of order $q$ with respect to the time $t$. We prove that the problem admits no global weak solution other than the trivial one with suitable initial data when $1<p<1+\frac{2\beta }{d\beta +2(1?\beta )}$. Next, we deal with the system

where $1<\alpha <2$, $0<\beta <1$, $p>1$, and $q>1$. We prove that the system admitsnon global weak solution other than the trivial one with suitable initial data when $1<pq<1+\frac{2\beta }{d\beta +2(1?\beta )}max\{p+1,q+1\}$. Our approach is based on the test function method.     

Pancyclic out-arcs of a vertex in oriented graphs

Let D be an oriented graph with $n?9$ vertices and minimum degree at least $n?2$, such that, for any two vertices x and y, either x dominates y or $d_D^{+}(x)+d_D^{?}(y)?n?3$. Song (1994) [5] proved that D is pancyclic. Bang-Jensen and Guo (1999) [2] proved, based on Song?s result, that D is vertex pancyclic. In this article, we give a sufficient condition for D to contain a vertex whose out-arcs are pancyclic in D, when $n?14$.     

Energy decay rate for a von Karman system with a boundary nonlinear delay term

In this paper, we show the energy decay rate for a von Karman system with a boundary nonlinear delay term. This work is devoted to investigate the influence of kernel function $g$ and the effect of the boundary nonlinear term ${\mu }_1{\left|u_t\left(t\right)\right|}^{m?1}{u}_t\left(t\right)$, a boundary nonlinear time delay term ${\mu }_2{\left|u_t(t?\tau )\right|}^{m?1}{u}_t(t?\tau )$ and prove energy decay rates of solutions when $g$ do not necessarily decay exponentially and the boundary condition has a time delay.     

Blow-up and decay for a class of pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity

We consider the pseudo-parabolic $p$-Laplacian equation $u_t?\Delta u_t?{\Delta }_{p}u={\left|u\right|}^{q?2}ulog\left(\left|u\right|\right)$ in a bounded domain with homogeneous Dirichlet boundary conditions. When $2<p<q<p(1+\frac{2}{n})$, we obtain results of the decay and the finite time blow-up for weak solutions.     

Boundedness in the higher-dimensional quasilinear chemotaxis-growth system with indirect attractant production

This paper deals with the following quasilinear chemotaxis-growth system

in a smoothly bounded domain $\Omega ?{\mathbb{R}}^n(n\ge 3)$ under zero-flux boundary conditions. The parameters $\mu ,\delta$ and $\tau$ are positive and the diffusion function $D\left(u\right)$ is supposed to generalize the prototype $D\left(u\right)\ge D_0{u}^{\theta }$ with $D_0>0$ and $\theta \in \mathbb{R}$. Under the assumption $\theta >1?\frac{4}{n}$, it is proved that whenever $\mu >0$, $\tau >0$ and $\delta >0$, for any given nonnegative and suitably smooth initial data ($u_0$, $v_0$, $w_0$) satisfying $u_0?0$, the corresponding initial–boundary problem possesses a unique global solution which is uniformly-in-time bounded. The novelty of the paper is that we use the boundedness of the $\left|\right|v(?,t)|{|}_{W^{1,s}\left(\Omega \right)}$ with $s\in [1,\frac{2n}{n?2})$ to estimate the boundedness of $\left|\right|?v(?,t)|{|}_{L^{2q}\left(\Omega \right)}(q>1)$. Moreover, the result in this paper can be regarded as an extension of a previous consequence on global existence of solutions by Hu et al. (2016) under the condition that $D\left(u\right)\equiv 1$ and $n=3$.