tailieunhanh - Quenching behavior of a semilinear reaction-diffusion system with singular boundary condition

In this paper, we study the quenching behavior of the solution of a semilinear reaction-diffusion system with singular boundary condition. We first get a local exisence result. Then we prove that the solution quenches only on the right boundary in finite time and the time derivative blows up at the quenching time under certain conditions. | Turk J Math (2016) 40: 166 – 180 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Quenching behavior of a semilinear reaction-diffusion system with singular boundary condition Burhan SELC ¸ UK∗ Department of Computer Engineering, Karab¨ uk University, Karab¨ uk, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we study the quenching behavior of the solution of a semilinear reaction-diffusion system with singular boundary condition. We first get a local exisence result. Then we prove that the solution quenches only on the right boundary in finite time and the time derivative blows up at the quenching time under certain conditions. Finally, we get lower bounds and upper bounds for quenching time. Key words: Reaction-diffusion system, singular boundary condition, quenching, maximum principles, monotone iterations 1. Introduction In this paper, we study the quenching behavior of the solution of the following semilinear reaction-diffusion system with singular boundary condition: −p ut = uxx + (1 − v) 1 , 0 0, v (x, 0) = v0 (x) > 0, 0 ≤ x ≤ 1, () where p1 , p2 , q1 , q2 are positive constants and u0 (x), v0 (x) are nonnegative smooth functions satisfying the compatibility conditions. They proved that the solution blows up only on the right boundary in finite time under certain conditions. Finally, they obtained the blow-up rate. The equivalence between the blow-up problem and the quenching problem is well known; for example, see [10] and [15]. Motivated by problems () and (), we investigate the quenching behavior of problem (). In Section 2, we give a local existence result for problem () . In Section 3, we prove that quenching occurs in finite time, the only quenching point is x = 1 , and (ut , vt ) blows up at quenching time under certain conditions. In Section 4, we obtain .

• Toán học
• Quenching behavior
• Reaction diffusion system
• Singular boundary condition
• Maximum principles
• Monotone iterations