论文标题
抛物线方程的有界斜率条件，具有时间相关的集成
The bounded slope condition for parabolic equations with time-dependent integrands
论文作者
论文摘要
在本文中，我们研究了cauchy-dirichlet问题\ begin {equation*} \左边\{ \ begin {array} {ll} \ mbox {$ \ partial_t u- \ operatorAtorname {div} \ left（d_ξf（t，du） \ mbox {$ u = u_o $}＆\ mbox {on $ \ partial _ {\ mathcal {p}}ω__t$}，\\ [5pt] \ end {array} \正确的。 \ end {qore*}其中$ω\ subset \ mathbb {r}^n $是一个convex域，$ f：[0，t] \ times \ times \ times \ mathbb {r}^n \ rightArrow \ rightArrow \ mathbb {r} $ in and convex in and and convex in and and convex in and and convex in and consects in and and convex in第二个变量。假设初始和边界基准$ u_o：\OverlineΩ\ rightarrow \ Mathbb {r} $满足有界的斜率条件，我们证明存在一个独特的变性解决方案，该解决方案是LipsChitz在太空变量中连续的。
In this paper, we study the Cauchy-Dirichlet problem \begin{equation*} \left\{ \begin{array}{ll} \mbox{$\partial_t u - \operatorname{div} \left( D_ξf(t, Du)\right) = 0$ } & \mbox{in $Ω_T$}, \\[5pt] \mbox{$u = u_o$} & \mbox{on $\partial_{\mathcal{P}} Ω_T$},\\[5pt] \end{array} \right. \end{equation*} where $Ω\subset \mathbb{R}^n$ is a convex domain, $f:[0,T]\times\mathbb{R}^n \rightarrow \mathbb{R}$ is $L^1$-integrable in time and convex in the second variable. Assuming that the initial and boundary datum $u_o:\overlineΩ\rightarrow \mathbb{R}$ satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.
