Xianfa Song

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.

Sharp thresholds of global existence and blowup for a system of Schrödinger equations with combined power-type nonlinearities

In this paper, we consider the Cauchy problem of a nonlinear Schrodinger system. Through establishing a sharp weighted vector-valued Gagliardo–Nirenberg’s inequality, we find that the best constant in this inequality can be regarded as the criterion of blowup and global existence of the solutions when p=4/N. And we prove that the solutions of this system will always exist globally if p<4/N. The sharp thresholds for blowup and global existence are also obtained when 4/N≤p<4/(N−2)+.

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.

Bounds for the blowup time and blowup rate estimates for a type of parabolic equations with weighted source

Abstract In this paper, we will consider the blowup phenomena for a type of parabolic equations with weighted nonlinear source. We obtain the estimates of the blowup rate and the bounds for blowup time of the solution to the problem in any smooth bounded domain Ω ⊂ R n ( n ⩾ 3 ) . In some special cases, we can even get the exact values of blowup rate and blowup time.

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.

Critical exponents in a parabolic system with inner absorption and coupled nonlinear boundary flux

This paper deals with critical exponents for a parabolic system with inner absorption and coupled nonlinear boundary flux. A detail analysis is given to the interactions among the multi-nonlinearities in the system. The critical exponent of the system is simply described via a characteristic matrix equation introduced.

The lower bound for the blowup time of the solution to a quasi-linear parabolic problem

Abstract In this paper, using a delicate application of general Sobolev inequality, we establish the lower bound for the blowup time of the solution to a quasi-linear parabolic problem, which improves the result of Theorem 2.1 in Bao and Song (2014).

ON GLOBAL ROUGH SOLUTIONS TO A NON-LINEAR SCHRÖDINGER SYSTEM

The non-linear Schrodinger systems arise from many important physical branches. In this paper, employing the ‘ I -method’, we prove the global-in-time well-posedness for a coupled non-linear Schrodinger system in H s ( n ) when n = 2, s > 4/7 and n = 3, s > 5/6, respectively, which extends the results of J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao (Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation, Math Res. Lett . 9 , 2002, 659–682) to the system.

The explicit solution to the initial–boundary value problem of Gierer–Meinhardt model

Abstract Using the solutions of an elliptic system and an ODE system, under certain conditions, we get the explicit solution to the following initial–boundary value problem of Gierer-Meinhardt model u t = d 1 Δ u − a 1 u + u p v q + δ 1 ( x , t ) , x ∈ Ω , t > 0 v t = d 2 Δ v − a 2 v + u r v s + δ 2 ( x , t ) , x ∈ Ω , t > 0 ∂ u ∂ η = ∂ v ∂ η = 0 ( or u = v = 0 ) , x ∈ ∂ Ω , t > 0 u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω . Here p > 1 , s > − 1 , d 1 , d 2 , q , r > 0 and a 1 , a 2 ≥ 0 are constants, while δ 1 ( x , t ) and δ 2 ( x , t ) are nonnegative continuous functions.