Asymptotic behavior of solutions to the logarithmic diffusion equation with a linear source
Asymptotic behavior of solutions to the logarithmic diffusion equation with a linear source
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DOI：
10.1007/s00208-017-1604-5
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发表时间：
2017-10
影响因子：
1.4
通讯作者：
Masahiko Shimojo;P. Takáč;E. Yanagida
中科院分区：
文献类型：
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作者：
Masahiko Shimojo;P. Takáč;E. Yanagida
我们研究柯西问题正解的行为，其中 α、β α、β 被给定正常数，u_0 (x) u 0 (x) 是正初始值。在质量守恒的情况下，即，对于 t ≥ 0 t≥ 0 ， ∫ _-∞^ ∞ u (x, t) dx == α+ β∫-Infinity u (x, t) dxeq α+ β ，我们通过交集数论证证明解逼近行波，即 t → ∞ t→∞。然后，我们通过变量变换来研究大规模膨胀或灭绝情况下的行为。当总质量较小时，我们表明解的消光发生在有限时间内，并且重新缩放的解收敛到行波，而当总质量较大时，解呈指数增长并且重新缩放的解收敛到某个轮廓。我们的结果还包括解的一些对数凹性特性。
We investigate the behavior of positive solutions to the Cauchy problem where α, β α, β are given positive constants and u_0 (x) u 0 (x) is a positive initial value. In the case of mass conservation, ie, ∫ _-∞^ ∞ u (x, t) dx ≡ α+ β∫-∞∞ u (x, t) dx≡ α+ β for t ≥ 0 t≥ 0, we show by an intersection number argument that the solution approaches a traveling wave as t → ∞ t→∞. We then study the behavior in the case of mass expansion or extinction by using a transformation of variables. When the total mass is smaller, we show that extinction of the solution occurs in finite time and a rescaled solution converges to the traveling wave, whereas when the total mass is larger, the solution grows exponentially and a rescaled solution converges to a certain profile. Our results also include some log-concavity properties of solutions.