李仲庆.一类具梯度项和 源的抛物方程有界弱解的存在性[J].数学研究及应用,2022,42(3):279~288
一类具梯度项和 源的抛物方程有界弱解的存在性
Bounded Weak Solutions to a Class of Parabolic Equations with Gradient Term and $L^r{(0,T;L^q(\Omega))}$ Sources
投稿时间:2021-04-16  修订日期:2021-06-21
DOI:10.3770/j.issn:2095-2651.2022.03.006
中文关键词:  抛物方程  低阶梯度项  估计  有界解
英文关键词:parabolic equations  lower order gradient term  $L^\infty$ estimate  bounded solutions
基金项目:国家自然科学基金(Grant No.11901131),贵州财经大学校级科研项目(Grant No.2019XYB08).
作者单位
李仲庆 贵州财经大学数学与统计学院, 贵州 贵阳 550025 
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中文摘要:
      我们考虑了一类原型为$$\begin{cases}u_t-\Delta u=\overrightarrow{b}(x,t)\cdot\nabla u+\gamma|\nabla u|^2-\text{div}{\overrightarrow{F}(x,t)}+f(x,t), &(x,t)\in \Omega_T,\\ u(x,t)=0,&(x,t)\in\Gamma_T,\\ u(x,0)=u_0(x), &x\in\Omega,\end{cases}$$的一类抛物方程. 其中, 函数$|\overrightarrow{b}(x,t)|^2,|\overrightarrow{F}(x,t)|^2,f(x,t)$位于空间$L^r{(0,T;L^q(\Omega))}$, $\gamma$是一个正常数. 在源项和梯度的系数项在空间$L^r{(0,T;L^q(\Omega))}$具有合适的可积条件下, 本文的目的在于证明先验的$L^\infty$估计以及方程存在有界解. 主要的方法包括通过正则化建立扰动问题, 用非线性的检验函数实现Stampacchia迭代技术以及极限过程中的紧性论断.
英文摘要:
      We consider a class of nonlinear parabolic equations whose prototype is $$\begin{cases}u_t-\Delta u=\overrightarrow{b}(x,t)\cdot\nabla u+\gamma|\nabla u|^2-\text{div}{\overrightarrow{F}(x,t)}+f(x,t), &(x,t)\in \Omega_T,\\ u(x,t)=0,&(x,t)\in\Gamma_T,\\ u(x,0)=u_0(x), &x\in\Omega, \end{cases}$$ where the functions $|\overrightarrow{b}(x,t)|^2,|\overrightarrow{F}(x,t)|^2,f(x,t)$ lie in the space $L^r{(0,T;L^q(\Omega))}$, $\gamma$ is a positive constant. The purpose of this paper is to prove, under suitable assumptions on the integrability of the space $L^r{(0,T;L^q(\Omega))}$ for the source terms and the coefficient of the gradient term, a priori $L^\infty$ estimate and the existence of bounded solutions. The methods consist of constructing a family of perturbation problems by regularization, Stampacchia's iterative technique fulfilled by an appropriate nonlinear test function and compactness argument for the limit process.
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