论文标题
$ c^{1，α} $适用于非标准增长的准抛物线方程
$C^{1,α}$ regularity for quasilinear parabolic equations with nonstandard growth
论文作者
论文摘要
在本文中，我们获得了$ c^{1，α} $估计的某些准线性抛物线方程的弱解，满足非标准生长条件，原型示例为$$ u_t - \ text {div}（| \ nabla u |^nabla u |^{p-2} {p-2} {p-2} \ nabla u + a（t） 0，$$ $$ u_t - \ text {div}（| \ nabla u |^{p（t）-2} \ nabla u）= 0。$$，假设先验的解决方案有限制梯度。我们建立在最近开发的缩放和涵盖参数的基础上，这使我们能够以统一的方式考虑奇异和退化的案例，并且对相位切换因子$ a（t）$和变量指数$ p（t）$的规律要求最小。此外，我们能够采用任何$ p \ leq q <\ infty $来获得所需的规律性。
In this paper, we obtain $C^{1,α}$ estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being $$u_t - \text{div} (|\nabla u|^{p-2} \nabla u + a(t)|\nabla u|^{q-2} \nabla u) = 0,$$ $$u_t - \text{div} (|\nabla u|^{p(t)-2} \nabla u) = 0.$$ under the assumption that the solutions a priori have bounded gradient. We build on the recently developed scaling and covering argument which allows us to consider the singular and degenerate cases in a uniform manner and with minimal regularity requirements on the phase switching factor $a(t)$ and the variable exponent $p(t)$. Moreover, we are able to take any $p \leq q < \infty$ to obtain the desired regularity.
