AI 中文总结
本文研究次临界双非线性抛物型方程组弱解的局部正则性，在特定可积性假设下建立解的定量界，并证明梯度的局部更高可积性。
AI 中文摘要
我们考虑有界时空柱体Ω_T=Ω×(0,T)⊂ℝ^{N+1}中形如∂_t(|u|^{q-1}u)-div(|Du|^{p-2}Du)=div(|F|^{p-2}F)的非齐次双非线性抛物型方程组，研究次临界范围p≤N(q+1)/(N+q+1)且0<q<(N+2)/(N-2)下弱解的局部正则性性质。在额外可积性假设|u|∈L_{loc}^r(Ω_T)下，我们建立了|u|的定量界，其中指数r满足λ_r=N(p-q-1)+pr>0；此外，在p=N(q+1)/(N+q+1)且(N+p)/(N-p)<q<(N+2)/(N-2)的范围内，当|u|∈L_{loc}^r(Ω_T)成立时，我们证明了|Du|的局部更高可积性。
英文摘要
We consider the inhomogeneous doubly nonlinear parabolic systems of the form \begin{equation*}\partial_t (|u|^{q-1}u)-\operatorname{div}(|Du|^{p-2}Du)=\operatorname{div}(|F|^{p-2}F)\end{equation*} in a bounded space-time cylinder $Ω_T=Ω\times(0,T)\subset \mathbb{R}^{N+1}$. We study the local regularity properties for weak solutions in the subcritical range $p\leq\frac{N(q+1)}{N+q+1}$ and $0<q<\frac{N+2}{N-2}$. Under an extra integrability assumption $|u|\in L_{\loc}^{\rr}(Ω_T)$, we establish a quantitative bound for $|u|$. Here, the exponent $\rr$ satisfies $\mathrmλ_{\rr}=N(p-q-1)+p\rr>0$. In addition, we prove local higher integrability of $|Du|$ in the range $p=\frac{N(q+1)}{N+q+1}$ and $\frac{N+p}{N-p}<q<\frac{N+2}{N-2}$, provided that $|u|\in L_{\loc}^{\rr}(Ω_T)$ holds.
CommentsThe proof of Lemma 4.7 is wrong and the range (1.5) is an empty set
