一类含梯度项的奇异抛物型方程弱解的存在性

张亚茹; 夏莉

doi:10.6054/j.jscnun.2023040

一类含梯度项的奇异抛物型方程弱解的存在性

doi: 10.6054/j.jscnun.2023040

张亚茹¹,
夏莉^{1, 2, ,}

1.
广东财经大学统计与数学学院, 广州 510320
2.
广东财经大学大数据与教育应用统计实验室, 广州 510320

基金项目:

国家自然科学基金项目 11971200

广东省教育厅科研项目[特色创新项目] 2018KTSCX069

广东省教育厅委托项目 0835-210Z33606691

详细信息

通讯作者:
夏莉, Email: xaleysherry@163.com

中图分类号: O175.26
计量
- 文章访问数: 10
- HTML全文浏览量: 0
- PDF下载量: 1
- 被引次数: 0
出版历程
- 收稿日期: 2021-02-13
- 网络出版日期: 2023-08-26
- 刊出日期: 2023-06-25

Existence of Weak Solutions for Some Singular Parabolic Equations with Gradient Terms

ZHANG Yaru¹,
XIA Li^{1, 2
, ,}

1.
School of Statistics and Mathematics, Guangdong University of Finance and Economic, Guangzhou 510320, China
2.
Big Data and Educational Statistics Application Laboratory, Guangdong University of Finance and Economic, Guangzhou 510320, China

摘要

摘要: 文章研究一类具有Dirichlet边界条件和初始条件的含梯度项奇异抛物型偏微分方程： $\left\{\begin{array}{l}y_t-y^{\prime \prime}-\frac{\kappa}{r} y^{\prime}+\lambda \frac{\left|y^{\prime}\right|^2}{y^m}=f(r, t) \quad(y \geqslant 0, (r, t) \in(0, 1) \times(0, T]), \\y(0, t)=y(1, t)=0 \ \ \ \ \ \quad(t \in(0, T]), \\y(r, 0)=\varphi(r) \quad(r \in(0, 1)), \end{array}\right.$ 其中, T>0, κ≥0, λ>0, m∈(0, 2)。由于含梯度的奇异抛物型方程中具有奇异项和非线性项, 故先利用抛物正则化方法将方程进行正则化, 再结合上下解方法, 证明了在不同假设条件下的该类方程非负弱解的存在性。最后, 证明了该方程的弱比较原理。
- 存在性 /
- 奇异抛物型方程 /
- 弱解
Abstract: A class of singular parabolic equations with gradient terms is studied in this paper, with Dirichlet boun-dary condition and initial condition, of the from $\left\{\begin{array}{l}y_t-y^{\prime \prime}-\frac{\kappa}{r} y^{\prime}+\lambda \frac{\left|y^{\prime}\right|^2}{y^m}=f(r, t) \quad(y \geqslant 0, (r, t) \in(0, 1) \times(0, T]), \\y(0, t)=y(1, t)=0 \ \ \ \ \ \quad(t \in(0, T]), \\y(r, 0)=\varphi(r) \quad(r \in(0, 1)), \end{array}\right.$ where T>0, κ≥0, λ>0, m∈(0, 2). Since singular parabolic equations containing gradient terms have singular terms and nonlinear terms, the parabolic regularization method is used to regularize the equation, and then combined with the sub-super solutions method, the existence of weak solutions of the equations under different assumptions is proved. Finally, the weak comparison principle of the equations is proved.
- existence /
- singular parabolic equation /
- weak solution

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参考文献(18)

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