一类带有对数非线性项的热方程的梯度爆破问题
Gradient Blow Up Problem for a Class of Heat Equations with Logarithmic Nonlinear Terms
DOI: 10.12677/aam.2025.144220, PDF,    科研立项经费支持
作者: 王俊伟, 张玲玲*：太原理工大学数学学院，山西 晋中
关键词: 热方程；对数非线性项；梯度爆破；上下界；Heat Equation； Logarithmic Nonlinear Terms； Gradient Blow Up； Upper and Lower Bounds
摘要: 我们考虑了具有一般对数非线性项的一维半线性热方程的梯度爆破问题，也就是方程解本身有界，但解的梯度会趋于无穷。通过尺度变换和抛物估计，得到了解梯度的一个上界和下界。最后给出了一个特殊例子来验证。
Abstract: We considered the gradient blow up problem of the one-dimensional semi-linear heat equation with general logarithmic nonlinear term, which the solution of the equation is bounded but the gradient of the solution becomes unbounded. By the rescaling method and parabolic estimates, the upper and lower bounds of gradient blow up rate are established. Furthermore, an example is given to illustrate.
文章引用：王俊伟, 张玲玲. 一类带有对数非线性项的热方程的梯度爆破问题[J]. 应用数学进展, 2025, 14(4): 968-980. https://doi.org/10.12677/aam.2025.144220

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