Abstract
This paper is concerned with the blow-up properties of Cauchy and Dirichlet problems of a coupled system of Reaction-Diffusion equations with gradient terms. The main goal is to study the influence of the gradient terms on the blow-up profile. Namely, under some conditions on this system, we consider the upper blow-up rate estimates for its blow-up solutions and for the gradients.
Highlights
With some restricted conditions on system (1), we show that the upper blow-up rate estimates for this solution and its gradients terms take the following forms: u (x, t) ≤ C1 (T − t)−α
Since the system (1) is uniformly parabolic and its equations have the same principle parts and F1, F2 ∈ C1(R × Rn), the growths of the nonlinearities in F1 and F2 with respect to the gradient terms are subquadratic; u0, V0 ∈ C2(Ω), and satisfying (5), it follows that the local existence and uniqueness of classical solution to the for system (1), where Ω = BR, with zero Dirichlet boundary conditions, are guaranteed by standard parabolic theory
(ii) The gradient terms are bounded for any t < T
Summary
The profile of blow-up solutions of (6) is similar to that of problem (8), where q < 2p/(p + 1) (see [12]), while if q > 2p/(p + 1), the gradient term causes more effect on the plow-up profile and it becomes more singular It has been proved in [4, 13, 14], that there are positive constants a and b, such that the upper and lower blow-up rate estimates for this equation, where < 2p/(p + 1), take the following form:. With some restricted conditions on system (1), we show that the upper blow-up rate estimates for this solution and its gradients terms take the following forms:. |∇V (x, t)| ≤ C2 (T − t)−(1+2β)/2 , where (x, t) ∈ Ω × (0, T) and C1, C2 > 0, α, β are given in (15)
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