Sining Zheng

Blow-up analysis for a quasilinear parabolic system with multi-coupled nonlinearities

This paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux. As the results of the interaction among the multi-coupled nonlinearities in the system, some appropriate conditions for global existence and global nonexistence of solutions are determined respectively.

Blow-up and blow-up rate for a reaction–diffusion model with multiple nonlinearities ☆

Abstract This paper deals with interactions among three kinds of nonlinear mechanisms: nonlinear diffusion, nonlinear reaction and nonlinear boundary flux in a parabolic model with multiple nonlinearities. The necessary and sufficient blow-up conditions are established together with blow-up rate estimates for the positive solutions of the problem.

A quasilinear reaction-diffusion system coupled via nonlocal sources

In this paper, we establish the critical exponent and the blow-up rate for a quasilinear reaction-diffusion system with nonlocal nonlinear sources and inner absorptions, subject to homogeneous Dirichlet conditions and nonnegative initial data. It is found that the critical exponent is determined by the interaction among all the six nonlinear exponents from all the three kinds of the nonlinearities. While the blow-up rate is independent of the nonlinear diffusion exponents due to the effects of the coupled nonlocal sources. Two kinds of characteristic algebraic systems are introduced to get simple descriptions for the critical exponent and the blow-up rate, respectively. We prove moreover that the blow-up could be global due to the nonlocality of the nonlinear sources.

Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions

In this paper, the blow-up rate for a nonlinear diffusion equation with a nonlinear boundary condition is established together with the necessary and sufficient blow-up conditions.

Blow-up estimates for system of heat equations coupled via nonlinear boundary flux

Abstract This paper deals with a system of heat equations coupled via nonlinear boundary flux. The precise blow-up rate estimates are established together with the blow-up set. It is observed that there is some quantitative relationship regarding the blow-up properties between the heat system with coupled nonlinear boundary flux terms and the corresponding reaction–diffusion system with the same nonlinear terms as the source.

Coexistence solutions for a reaction-diffusion system of un-stirred chemostat model

In this paper, we study a reaction-diffusion system of multiple food chain model, where two predators feed on a single prey growing in an un-stirred chemostat. The conditions for the coexistence of steady states are determined. The main technique used here is the degree theory in cones.

Asymptotic behavior for a reaction-diffusion equation with inner absorption and boundary flux ☆

Abstract This paper deals with a reaction-diffusion equation with inner absorption and boundary flux of exponential forms. The blow-up rate is determined with the blow-up set, and the blow-up profile near the blow-up time is obtained by the Giga–Kohn method. It is observed that the blow-up rate and profile are independent of the nonlinear absorption term.

Non-simultaneous blow-up in a reaction-diffusion system

This paper deals with non-simultaneous blow-up for a reaction-diffusion system with absorptions and coupled non-linear boundary flux. We establish the necessary and sufficient conditions for the occurrence of non-simultaneous blow-up with proper initial data. Moreover, we determine the conditions under which any blow-up would be non-simultaneous.

Multinonlinear interactions in quasi-linear reaction-diffusion equations with nonlinear boundary flux

This paper deals with multinonlinearity interactions for more general quasi-linear reaction-diffusion models, where three nonlinear mechanisms-nonlinear diffusion, nonlinear reaction (or absorption), and nonlinear boundary flux-are included. The absorption/ boundary flux balance is studied to get the global solvability conditions as well as the blow-up criterion for the model with boundary flux and absorption. Specifically, the global solvability conditions established in this paper for the model with boundary flux and reaction are both necessary and sufficient.

Second critical exponent for evolution p-Laplacian equation with weighted source☆

This paper studies the Cauchy problem for the evolution p-Laplacian equation with weighted source: ut=div(|∇u|p−2∇u)+|x|muq in Rn×(0,T), where 2n/(n+1) n(2−p)−p, q>1. We obtain the second critical exponent on decay orders of initial data at infinity to identify global and non-global solutions in the coexistence region of the two kinds of solutions. In addition, we determine the life span of the non-global solutions in the case of m=0.