In this paper we study the regularity and the behavior in time of the solutions to a quasilinear class of noncoercive problems whose prototype is

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t} - \mathrm{div}(a(x,t,u)\nabla u) = -\mathrm{div}(u\,E(x,t)) &{}\quad \text{ in }\, \Omega \times (0,T), \\ u(x,t) = 0 &{}\quad \text{ on }\, \partial \Omega \times (0,T), \\ u(x,0) = u_{0}(x) &{}\quad \text{ in }\, \Omega . \end{array} \right. \end{aligned}$$

In particular we show that under suitable conditions on the vector field E, even if the problem is noncoercive and although the initial datum \(u_0\) is only an \(L^{1}(\Omega )\) function, there exist solutions that immediately improve their regularity and belong to every Lebesgue space. We also prove that solutions may become immediately bounded. Finally, we study the behavior in time of such regular solutions and we prove estimates that allow to describe their blow-up for t near zero.

中文翻译：

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非强制抛物线问题的正则结果和渐近行为

在本文中，我们研究了一类拟线性非强制问题的解的规律性和时间行为，其原型为

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t} - \mathrm{div}(a(x,t,u)\nabla u) = -\mathrm{div}( u\,E(x,t)) &{}\quad \text{ in }\, \Omega \times (0,T), \\ u(x,t) = 0 &{}\quad \text{在 }\, \partial \Omega \times (0,T), \\ u(x,0) = u_{0}(x) &{}\quad \text{ in }\, \Omega 上。\end{数组} \对。\end{对齐}$$

特别是我们表明，在向量场E的合适条件下，即使问题是非强制的，虽然初始数据\(u_0\)只是一个\(L^{1}(\Omega )\)函数，但存在立即改善其规律性并属于每个 Lebesgue 空间的解决方案。我们还证明了解决方案可能会立即成为有界的。最后，我们研究了这种常规解决方案的时间行为，并证明了可以描述它们在t接近零时的爆炸的估计。