Abstract
This paper studies quenching properties of solutions of a semilinear parabolic system with localized reaction sources in a square domain. The system has the homogeneous Dirichlet boundary condition and null initial condition. We prove that solutions quench simultaneously, and compute approximated critical values of the system using a numerical method.
Highlights
This paper studies quenching properties of solutions of a semilinear parabolic system with localized reaction sources in a square domain
We prove that solutions quench simultaneously, and compute approximated critical values of the system using a numerical method
We study the following semilinear parabolic system:
Summary
Let β1 ( x, y,t ) and β2 ( x, y,t ) be nontrivial, nonnegative, and bounded functions on D × (0, ∞). Suppose that u ( x, y,t ) and v ( x, y,t ) are solutions of the following system: Lu ≥ β1 ( x, y,t ) v (0, 0,t ) in D × (0, Γ), Lv ≥ β2 ( x, y,t )u (0, 0,t ) in D × (0, Γ), u ( x, y, 0) ≥ 0 and v ( x, y, 0) ≥ 0 on D, u ( x, y,t ) ≥ 0 and v ( x, y,t ) ≥ 0 on ∂D × (0, Γ). There exist some t ∈ (0, Γ) and ( x2 , y2 ) ∈ D such that ( ) ( ) Ψ x2 , y2 ,t = 0 , Ψt x2 , y2 ,t ≤ 0 , and Ψ attains its local minimum at ( x2 , y2 ,t). By Lemma 2.1, 0 is a lower solution of the problem (1.1)-(1.2). The existence of classical solutions of the problem (1.1)-(1.2) is able to obtain by the Schauder fixed point theorem of [[9], pp. By the maximum principle [[10], p. 54], ut > 0 and vt > 0 in D × (0, Γ)
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