Existence, uniqueness and asymptotic behavior of solutions for a singular parabolic equation
Existence, uniqueness and asymptotic behavior of solutions for a singular parabolic equation
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DOI：
10.1016/j.jmaa.2009.04.039
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发表时间：
2009-10
影响因子：
1.3
通讯作者：
Li Xia;Z. Yao
中科院分区：
文献类型：
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作者：
Li Xia;Z. Yao
在本文中，我们关注一个奇异抛物线方程∂v∂t−Δv=f(x,t)−μ|∇v|2v在光滑有界域Ω∧rn服从零狄利克雷边界条件和初始条件φ大于或等于0。在μ， φ和f（x,t）的假设下，应用抛物正则化方法和次超解方法得到了一些存在唯一性结果。我们还讨论了解在L2（0，T;W01,2（Ω））和L∞（0，T;L2（Ω））范数为μ→0或μ→∞意义上的渐近行为。作为一个副产品，我们得到了一些在边界上爆炸的问题的解的存在性。
In this paper, we are concerned with a singular parabolic equation ∂v∂t−Δv=f(x,t)−μ|∇v|2v in a smooth bounded domain Ω⊂RNsubject to zero Dirichlet boundary condition and initial condition φ⩾0. Under the assumptions on μ, φ and f(x,t), some existence and uniqueness results are obtained by applying parabolic regularization method and the sub-supersolutions method. We also discuss the asymptotic behaviors of solutions in the sense of L2(0,T;W01,2(Ω)) and L∞(0,T;L2(Ω)) norms as μ→0 or μ→∞. As a byproduct we obtain the existence of solutions for some problems which blow up on the boundary.