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A remark on the note: “Partial Hölder continuity of the spatial derivatives of the solutions to nonlinear parabolic systems with quadratic growth”. (English) Zbl 0990.35062
The authors prove the partial regularity of the first spatial derivatives of the weak solutions of the parabolic system $u_t-\text{div} A(t,x,u,Du)=B^0(t,x,u,Du)$ with $$B^0(t,x,u,p)$$ growing quadratically with respect to $$p$$, $$A(t,x,u,p)$$, $$A_x$$ and $$A_u$$ growing linearly with respect to $$p$$ and $$A_p$$ bounded. In addition, it is supposed that $$A\in C^1(\overline Q\times \mathbb{R}^N\times \mathbb{R}^{nN})$$ and $$A_p$$ are uniformly continuous. The theorem on partial regularity of $$Du$$ is established here under the weaker assumptions on the solution $$u$$, namely, $$u\in L^2(-T,0;H^2(\Omega))\cap C^{0,\gamma}(\overline Q)$$. By means of an interpolation inequality it is proved that $$Du\in L^4(Q)$$. Using this information, the fact that $$u_t\in L^2(Q)$$ can be deduced from the system. Having these ingredient, one can finish the proof as in the former paper mentioned in the title [ibid. 76, 219-245 (1986; Zbl 0622.35030)].
Reviewer: O.John (Praha)

##### MSC:
 35K55 Nonlinear parabolic equations 35B65 Smoothness and regularity of solutions to PDEs
Full Text:
##### References:
 [1] M. Marino - A. Maugeri , Partial Hölder continuity of the spatial derivatives of the solutions to nonlinear parabolic systems with quadratic growth , Rend. Sem. Mat. Padova , 76 ( 1986 ), pp. 219 - 245 . Numdam | MR 881572 | Zbl 0622.35030 · Zbl 0622.35030 [2] C. Miranda , Su alcuni teoremi di inclusione , Annales Polonici Math. , 16 ( 1965 ), pp. 305 - 315 . MR 187077 | Zbl 0172.40303 · Zbl 0172.40303 [3] L. Nirenberg , An extended interpolation inequality , Ann. Sc. Norm. Sup. Pisa ( 3 ), 20 ( 1966 ), pp. 733 - 737 . Numdam | MR 208360 | Zbl 0163.29905 · Zbl 0163.29905
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