In this paper, we study the quenching behavior of the solution of a semilinear heat equation with a singular boundary outflux. We prove a finite-time quenching for the solution. Further, we show that quenching occurs on the boundary under certain conditions and we show that the time derivative blows up at a quenching point. Finally, we get a quenching rate and a lower bound for the quenching time.Theorem 4. If u 0 satisfies (2), (3), (4) and (5), then there exists a positive constant C 1 such thatfor t sufficiently close to T .since u x < 0, J(x, t) cannot attain a positive interior maximum. On the other hand, J(x, 0) ≤ 0 by (4) and J(0, t) = 0, J(1, t) = 0, for t ∈ (0, T ). By the maximum principle, we obtain that J(Integrating for t from t to T we getwhere C 1 = (p + 1) 1/(p+1) . Remark 3. We can calculate a lower bound for the quenching time. From Theorem 4, a lower bound is (1 − u 0 (0)) p+1 /(p + 1) for quenching time T . If we choose, as in Remark 1, u 0 (x) = 0.9 − 2 3 x 4.5 , then we have T = 10 −11 for p = 9.
In this paper, we study the quenching behavior of solution of a nonlinear parabolic equation with a singular boundary condition. We prove finite-time quenching for the solution. Further, we show that quenching occurs on the boundary under certain conditions. Furthermore, we show that the time derivative blows up at quenching point. Also, we get a lower solution and an upper bound for quenching time. Finally, we get a quenching rate and lower bounds for quenching time.
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.
a b s t r a c tHypercube is a popular and more attractive interconnection networks. The attractive properties of hypercube caused the derivation of more variants of hypercube. In this paper, we have proposed two variants of hypercube which was called as "Fractal Cubic Network Graphs", and we have investigated the Hamiltonian-like properties of Fractal Cubic Network Graphs FCNG r ðnÞ. Firstly, Fractal Cubic Network Graphs FCNG r ðnÞ are defined by a fractal structure. Further, we show the construction and characteristics analyses of FCNG r ðnÞ where r ¼ 1 or r ¼ 2. Therefore, FCNG r ðnÞ is a Hamiltonian graph which is obtained by using Gray Code for r ¼ 2 and FCNG 1 ðnÞ is not a Hamiltonian Graph. Furthermore, we have obtained a recursive algorithm which is used to label the nodes of FCNG 2 ðnÞ. Finally, we get routing algorithms on FCNG 2 ðnÞ by utilizing routing algorithms on the hypercubes.
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