Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-08-20 , DOI: 10.1007/s00526-020-01810-9 Amal Attouchi , Philippe Souplet
Consider the diffusive Hamilton–Jacobi equation
$$\begin{aligned} u_t-\Delta u=|\nabla u|^p+h(x)\ \ \text { in } \Omega \times (0,T) \end{aligned}$$with Dirichlet conditions, which arises in stochastic control problems as well as in KPZ type models. We study the question of the gradient blowup rate for classical solutions with \(p>2\). We first consider the case of time-increasing solutions. For such solutions, the precise rate was obtained by Guo and Hu (2008) in one space dimension, but the higher dimensional case has remained an open question (except for radially symmetric solutions in a ball). Here, we partially answer this question by establishing the optimal estimate
$$\begin{aligned} C_1(T-t)^{-1/(p-2)}\le \Vert \nabla u(t)\Vert _\infty \le C_2(T-t)^{-1/(p-2)} \end{aligned}$$(1)for time-increasing gradient blowup solutions in any convex, smooth bounded domain \(\Omega \) with \(2<p<3\). We also cover the case of (nonradial) solutions in a ball for \(p=3\). Moreover we obtain the almost sharp rate in general (nonconvex) domains for \(2<p\le 3\). The proofs rely on suitable auxiliary functionals, combined with the following, new Bernstein-type gradient estimate with sharp constant:
$$\begin{aligned} |\nabla u|\le d_\Omega ^{-1/(p-1)}\bigl (d_p+C d_\Omega ^\alpha \bigr ) \ \ \text { in } \Omega \times (0,T),\qquad d_p=(p-1)^{-1/(p-1)}, \end{aligned}$$(2)where \(d_\Omega \) is the function distance to the boundary. This close connection between the temporal and spatial estimates (1) and (2) seems to be a completely new observation. Next, for any \(p>2\), we show that more singular rates may occur for solutions which are not time-increasing. Namely, for a suitable class of solutions in one space-dimension, we prove the lower estimate \(\Vert u_x(t)\Vert _\infty \ge C(T-t)^{-2/(p-2)}\).
中文翻译：
弥散性Hamilton-Jacobi方程的梯度爆炸率和陡峭梯度估计
考虑扩散哈密顿–雅各比方程
$$ \ begin {aligned} u_t- \ Delta u = | \ nabla u | ^ p + h（x）\ \ \ text {in} \ Omega \ times（0，T）\ end {aligned} $$在随机控制问题以及KPZ型模型中都会出现Dirichlet条件。我们用\（p> 2 \）研究经典解的梯度爆破率问题 。我们首先考虑耗时的解决方案。对于这样的解决方案，Guo和Hu（2008）在一个空间维度上获得了精确的比率，但是高维情况仍然是一个悬而未决的问题（球中径向对称的解决方案除外）。在这里，我们通过建立最佳估计来部分回答这个问题
$$ \ begin {aligned} C_1（Tt）^ {-1 /（p-2）} \ le \ Vert \ nabla u（t）\ Vert _ \ infty \ le C_2（Tt）^ {-1 /（p -2）} \ end {aligned} $$（1）在任何带有\（2 <p <3 \）的凸，光滑有界域\（\ Omega \）中用于时间增加的梯度爆破解。我们还讨论了\（p = 3 \）在球中的（非径向）解的情况。此外，我们在\（2 <p \ le 3 \）的一般（非凸）域中获得了几乎较快的比率。证明依赖于合适的辅助功能，并结合以下具有尖锐常数的新的Bernstein型梯度估计：
$$ \ begin {aligned} | \ nabla u | \ le d_ \ Omega ^ {-1 /（p-1）} \ bigl（d_p + C d_ \ Omega ^ \ alpha \ bigr）\ \ \ text {in} \ Omega \ times（0，T），\ qquad d_p =（p-1）^ {-1 /（p-1）}，\ end {aligned} $$（2）其中\（d_ \ Omega \）是到边界的函数距离。时间和空间估计（1）和（2）之间的这种紧密联系似乎是一个全新的观察。接下来，对于任何\（p> 2 \），我们表明对于不随时间增加的解决方案，可能会出现更多的奇异率。也就是说，对于在一维空间中的一类合适的解决方案，我们证明了较低的估计\（\ Vert u_x（t）\ Vert _ \ infty \ ge C（Tt）^ {-2 /（p-2）} \ ）。
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