This paper deals with a coupled system of fourth-order parabolic inequalities ｜u｜t ≥ -△2^u＋｜v｜^q, ｜v｜t ≥-△2v＋｜u｜p^ in S=R^n ×R^＋ withp, q 〉 1, n ≥1. AFujita- Liouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n/4≤ max（ p＋1/pq-1, q＋1/pq-1 ）. Since the general maximum-comparison principle does not hold for the fourth-order problem, the authors use the test function method to get the global non-existence of nontrivial solutions.
This paper deals with the quenching behavior of positive solutions to the Newton filtration equations coupled with boundary singularities.We determine quenching rates for non-simultaneous quenching at first,and then establish the criteria to identify the simultaneous and non-simultaneous quenching in terms of the parameters involved.
This article deals with a nonlocal heat system subject to null Dirichlet bound- ary conditions, where the coupling nonlocal sources consist of mixed type asymmetric non- linearities. We at first give the criterion for simultaneous blow-up of solutions, and then establish the uniform blow-up profiles of solutions near the blow-up time. It is observed that not only the simultaneous blow-up rates of the two components u and v are asymmet- ric, but also the blow-up rates of the same component u （or v） may be in different levels under different dominations.
In this paper, we investigate the blow-up properties of a quasilinear reaction-diffusion system with nonlocal nonlinear sources and weighted nonlocal Dirichlet boundary conditions. The critical exponent is determined under various situations of the weight functions. It is observed that the boundary weight functions play an important role in determining the blow-up conditions. In addition, the blow-up rate estimate of non-global solutions for a class of weight functions is also obtained, which is found to be independent of nonlinear diffusion parameters m and n.
This paper studies heat equation with variable exponent ut = △u ＋ Up（x） 4- Uq in RN × （0, T）, where p（x） is a nonnegative continuous, bounded function, 0 〈 p- = infp（x） ≤ p（x） ≤ supp（x） = p＋. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max{p＋,q} ≤1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 〈 q ≤ 1 with p＋ 〉 1, or 1 〈 q 〈 1 ＋2/N. In addition, if q 〉 1 ＋2/N, then （i） all solutions blow up in finite time with 0 〈 p- ≤ p＋ ≤ 1 ＋2/N; （ii） there are both global and nonglobal solutions for p- ≤ 1 ＋ 2/N; and （iii） there are functions p（x） such that all solutions blow up in finite time, and also functions p（x） such that the problem possesses global solutions when p＋〈 1＋2/N 〈 p＋.
