张亚茹, 夏莉. 一类含梯度项的奇异抛物型方程弱解的存在性[J]. 华南师范大学学报(自然科学版), 2023, 55(3): 96-102. doi: 10.6054/j.jscnun.2023040
引用本文: 张亚茹, 夏莉. 一类含梯度项的奇异抛物型方程弱解的存在性[J]. 华南师范大学学报(自然科学版), 2023, 55(3): 96-102. doi: 10.6054/j.jscnun.2023040
ZHANG Yaru, XIA Li. Existence of Weak Solutions for Some Singular Parabolic Equations with Gradient Terms[J]. Journal of South China Normal University (Natural Science Edition), 2023, 55(3): 96-102. doi: 10.6054/j.jscnun.2023040
Citation: ZHANG Yaru, XIA Li. Existence of Weak Solutions for Some Singular Parabolic Equations with Gradient Terms[J]. Journal of South China Normal University (Natural Science Edition), 2023, 55(3): 96-102. 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

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

  • 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\\beginarrayly_t-y^\prime \prime-\frac\kappar y^\prime+\lambda \frac\left|y^\prime\right|^2y^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)), \endarray\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。

     

    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\\beginarrayly_t-y^\prime \prime-\frac\kappar y^\prime+\lambda \frac\left|y^\prime\right|^2y^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)), \endarray\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.

     

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