具对数源项的p-Laplace方程解的整体存在性和爆破性

发布时间：2019-03-23 21:00

【摘要】：本文主要研究具非线性对数源项和p-Laplace算子的抛物问题解的整体存在性与爆破性,即考虑如下问题首先给出预备知识和主要结果,其次利用位势井方法以及能量估计,Sobolev嵌入不等式和反证法等证明解的整体存在性和解的正无穷时刻爆破性.具体讲,根据初始能量和M=1/p2(p2e/n(?)p)n/p的大小关系以及I(u_0)=∫Ω|%絬_0|pdx-∫Ω|u_0|plog|u_0|dx的非负性,主要结论如下:定理1.若u_0(x)∈ W_0~(1,p)(Ω),J(u_0)M,I(u_0)≥ 0,则问题(0.1)有一个整体弱解u ∈ L~∞(0,+∞;w_0~(1,p)(Ω)),u,∈ L~2(0,+∞;L~2(Ω)).进一步,对所有t≥ 0,有如下估计定理2.若u_0(x)∈ W_0~(1,p)(Ω),J(u_0)= M,I(u_0)≥ 0,则问题(0.1)有一个整体弱解u ∈ L~∞(0,+∞;W_0~(1,p)(Ω)),u,∈ L~2(0,+∞;L~2(Ω)).进一步,若 I(u_0)0,对任意给定的正数γ,都存在t0,使得对所有Ω t,都有定理3.若 u_0(x)∈ W_0~(1,p)(Ω),J(u_0)≤ M,I(u_0)0,则问题(0.1)的解 u = u(x,t)在正无穷时刻爆破,且有
[Abstract]:In this paper, we mainly study the global existence and blow-up of solutions to parabolic problems with nonlinear logarithmic source terms and p-Laplace operators, that is, considering the following problems, we first give the preparatory knowledge and the main results, secondly, we use the potential well method and the energy estimation, Sobolev's embedding inequality and counterproof are used to prove the global existence of solutions and the blow-up of positive infinity time. Specifically, based on the relationship between the initial energy and M=1/p2 (p2e/n (?) p) n) and the nonnegativity of I (u?) =? 惟 |%) _ 0 | pdx-? 惟 | u? If u _ 0 (x) 鈭，

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