Blow‐up of smooth solutions to the Navier–Stokes–Poisson equations

The main purpose of this paper is concerned with blow‐up smooth solutions to Navier–Stokes–Poisson (N‐S‐P) equations. First, we present a sufficient condition on the blow up of smooth solutions to the N‐S‐P system. Then we construct a family of analytical solutions that blow up in finite time. Copyright © 2010 John Wiley & Sons, Ltd.     

Blow‐up profiles of solutions for the exponential reaction‐diffusion equation

We consider the blow‐up of solutions for a semilinear reaction‐diffusion equation with exponential reaction term. It is known that certain solutions that can be continued beyond the blow‐up time possess a non‐constant self‐similar blow‐up profile. Our aim is to find the final time blow‐up profile for such solutions. The proof is based on general ideas using semigroup estimates. The same approach works also for the power nonlinearity. Copyright © 2011 John Wiley & Sons, Ltd.     

On a two-component Degasperis–Procesi shallow water system

We consider a two-component Degasperis–Procesi system which arises in shallow water theory. We analyze some aspects of blow up mechanism, traveling wave solutions and the persistence properties. Firstly, we discuss the local well-posedness and blow up criterion; a new blow up criterion for this system with the initial odd condition will be established. Finally, the persistence properties of strong solutions will also be investigated.     

Blow‐up criterion for two‐dimensional magneto‐micropolar fluid equations with partial viscosity

In this paper, we derive a blow‐up criterion of smooth solutions to the incompressible magneto‐micropolar fluid equations with partial viscosity in two space dimensions. Our proof is based on careful Hölder estimates of heat and transport equations and the standard Littlewood–Paley theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Blow up criterion for three‐dimensional nematic liquid crystal flows with partial viscosity

In this paper, we investigate the Cauchy problem for the three‐dimensional nematic liquid crystal flows with partial viscosity, and a blow up criterion of smooth solutions is established. This result is analogous to the celebrated Beale‐Kato‐Majda breakdown criterion for the incompressible Euler equations. Copyright © 2012 John Wiley & Sons, Ltd.     

Time‐weighted blow‐up rates and pointwise profile for single‐point blow‐up solutions in reaction–diffusion equations

This paper deals with asymptotic behavior for blow‐up solutions to time‐weighted reaction–diffusion equations u_t=Δu+e^αtv^p and v_t=Δv+e^βtu^q, subject to homogeneous Dirichlet boundary. The time‐weighted blow‐up rates are defined and obtained by ways of the scaling or auxiliary‐function methods for all α, . Aiding by key inequalities between components of solutions, we give lower pointwise blow‐up profiles for single‐point blow‐up solutions. We also study the solutions of the system with variable exponents instead of constant ones, where blow‐up rates and new blow‐up versus global existence criteria are obtained. Time‐weighted functions influence critical Fujita exponent, critical Fujita coefficient and formulae of blow‐up rates, but they do not limit the order of time‐weighted blow‐up rates and pointwise profile near blow‐up time. Copyright © 2017 John Wiley & Sons, Ltd.     

Blow‐up phenomena for a pseudo‐parabolic equation

By the means of a differential inequality technique, we obtain a lower bound for blow‐up time if p and the initial value satisfy some conditions. Also, we establish a blow‐up criterion and an upper bound for blow‐up time under some conditions as well as a nonblow‐up and exponential decay under some other conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Blow‐up phenomena in chemotaxis systems with a source term

This paper deals with a parabolic–parabolic Keller–Segel‐type system in a bounded domain of , {N = 2;3}, under different boundary conditions, with time‐dependent coefficients and a positive source term. The solutions may blow up in finite time t^?; and under appropriate assumptions on data, explicit lower bounds for blow‐up time are obtained when blow up occurs. Copyright © 2015 John Wiley & Sons, Ltd.     

Global solutions and blow‐up problems to a porous medium system with nonlocal sources and nonlocal boundary conditions

This paper deals with a porous medium system with nonlocal sources and weighted nonlocal boundary conditions. The main aim of this paper is to study how the reaction terms, the diffusion terms, and the weight functions in the boundary conditions affect the global and blow‐up properties to a porous medium system. The conditions on the global existence and blow‐up in finite time for nonnegative solutions are given. Furthermore, the blow‐up rate estimates of the blow‐up solutions are also established. Copyright © 2011 John Wiley & Sons, Ltd.     

Some new nonlinear wave solutions and their convergence for the (2+1)‐dimensional Broer–Kau–Kupershmidt equation

We use the bifurcation method of dynamical systems to study the (2+1)‐dimensional Broer–Kau–Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow‐up wave solutions, periodic smooth wave solutions, periodic blow‐up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow‐up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd.