This paper estimates the blow-up time for the heat equation u t = u with a local nonlinear Neumann boundary condition: The normal derivative ∂u/∂n = u q on 1 , one piece of the boundary, while on the rest part of the boundary, ∂u/∂n = 0. The motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects. We show the finite time blow-up of the solution and estimate both upper and lower bounds of the blow-up time in terms of the area of 1 . In many other work, they need the convexity of the domain and only consider the problem with 1 = ∂ . In this paper, we remove the convexity condition and only require ∂ to be C 2 . In addition, we deal with the local nonlinearity, namely 1 can be just part of ∂ .
This paper studies the heat equation ut = ∆u in a bounded domain Ω ⊂ R n (n ≥ 2) with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative ∂u/∂n = u q on partial boundary Γ1 ⊆ ∂Ω for some q > 1, while ∂u/∂n = 0 on the other part. We investigate the lower bound of the blow-up time T * of u in several aspects. First, T * is proved to be at least of order (q − 1) −1 as q → 1 + . Since the existing upper bound is of order (q − 1) −1 , this result is sharp. Secondly, if Ω is convex and |Γ1| denotes the surface area of Γ1, then T * is shown to be at least of order |Γ1| − 1 n−1 for n ≥ 3 and |Γ1| −1 ln |Γ1| −1 for n = 2 as |Γ1| → 0, while the previous result is |Γ1| −α for any α < 1 n−1 . Finally, we generalize the results for convex domains to the domains with only local convexity near Γ1. *
This paper studies the lower bound of the lifespan T * for the heat equation ut = ∆u in a bounded domain Ω ⊂ R n (n ≥ 2) with positive initial data u0 and a nonlinear radiation condition on partial boundary: the normal derivative ∂u/∂n = u q on Γ1 ⊆ ∂Ω for some q > 1, while ∂u/∂n = 0 on the other part of the boundary. Previously, under the convexity assumption of Ω, the asymptotic behaviors of T * on the maximum M0 of u0 and the surface area |Γ1| of Γ1 were explored. In this paper, without the convexity requirement of Ω, we will show that as M0 → 0 + , T * is at least of order M −(q−1) 0 which is optimal. Meanwhile, we will also prove that as |Γ1| → 0 + , T * is at least of order |Γ1| − 1 n−1 for n ≥ 3 and |Γ1| −1 ln |Γ1| −1 for n = 2. The order on |Γ1| when n = 2 is almost optimal. The proofs are carried out by analyzing the representation formula of u in terms of the Neumann heat kernel. *
This paper deals with the global dynamics for a SLIS epidemic model with infection age. In our model, we also consider the time delay in the progression from the latent individuals to becoming infectious individuals. We verify the well posedness of the model by changing it into an abstract nondensely defined Cauchy problem and find conditions for the existence of disease free equilibrium and endemic equilibrium. The theoretic analysis shows that the disease-free equilibrium is globally asymptotically stable as the basic reproduction number $R_{0}$R0 is less than unity and that the endemic equilibrium is locally asymptotically stable and the system is uniformly persistent as $R_{0}$R0 is greater than unity. The numerical simulations illustrate that the endemic equilibrium may be asymptotically stable as $R_{0}>1$R0>1.
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