We study the relationship of viscosity and weak solutions to the equation

$$\begin{aligned} \smash {\partial _{t}u-\varDelta _{p}u=f(Du)}, \end{aligned}$$

where \(p>1\) and \(f\in C({\mathbb {R}}^{N})\) satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when \(p\ge 2\).

中文翻译：

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ap-抛物线方程的粘性和弱解的等价性

我们研究了粘度和方程的弱解的关系

$$ \ begin {aligned} \ smash {\ partial _ {t} u- \ varDelta _ {p} u = f（Du）}，\ end {aligned} $$

其中\（p> 1 \）和\（f \ in C（{\ mathbb {R}} ^ {N}）\）满足适当的假设。我们的主要结果是，有界粘度上溶液与有界下半连续弱上层溶液一致。此外，我们证明了当（（p \ ge 2 \）时，弱上解的下半连续性。