This paper is concerned with the regularity of solutions to parabolic evolution equations. We consider semilinear problems on non-convex domains. Special attention is paid to the smoothness in the specific scale $B^r_{\tau,\tau}$, $\frac{1}{\tau}=\frac rd+ \frac 1p$ of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our proofs are based on Schauder's fixed point theorem.

翻译：本文研究抛物型演化方程解的正则性。我们考虑非凸域上的半线性问题，特别关注Besov空间特定尺度$B^r_{\tau,\tau}$（其中$\frac{1}{\tau}=\frac rd+ \frac 1p$）中的光滑性。这些空间中的正则性决定了自适应及其他非线性逼近格式所能达到的逼近阶。我们证明，在所有考虑的情形下，Besov正则性均足够高，足以证明自适应算法的使用是合理的。我们的证明基于Schauder不动点定理。