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This paper considers the singularity properties of positive solutions for a reaction-diffusion system with nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we establish the blow-up rate estimate for the blow-up solution.

In this paper, we use the constructive methods to show that every Schwartz function can be decomposed as a product of two Schwartz functions. The same is true for D(R n) function with the same compact set. As their applications, it immediately follows that S (R n) = S (R n)S (R n) and D(R n) = D(R n)D(R n). It should be pointed out that we use the adaptive idea and method to give the decompositions.

Abstract. This paper deals with blow-up properties of solutions to a semilinear parabolic system with nonlinear localized source involved a product with local termswith homogeneous Dirichlet boundary conditions. We investigate the influence of localized sources and local terms on blow-up properties for this system, and prove that: (i) when m, q 0 this system possesses uniform blow-up profiles, in other words, the localized terms play a leading role in the blow-up profile for this case; (ii) when m, q > 0 , this system presents single point blow-up patterns, or say that local terms dominate localized terms in the blow-up profile. Moreover, the blow-up rate estimates in time and space are obtained, respectively.

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