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Regularity in Sobolev spaces for doubly nonlinear parabolic equations. (English) Zbl 1029.35142
Let $$m>1$$ and $$p>(m+1)/m$$ be constants, and define $$A(\xi)=|\xi|^{p-2}\xi$$ and $$a(z)=|z|^{m-1}z$$. The authors examine the regularity of solutions of the Cauchy problem $u_{t} =\operatorname {div} A(D[a(u)])\quad \text{in } \mathbb R^{n}\times(0,T), \qquad u(x,0)=u_0(x) \quad\text{in } \mathbb R^{n}$ for $$n \geq 2$$ and $$T>0$$. If $$q<\max\{p/(p+1),2/3\}$$ and $$r<1/q$$, it is shown that $a(u) \in L^{q}(0,T;W^{r,q}(\mathbb R^{n})).$ Similar estimates are proved for powers of $$|Da(u)|$$. The proof is based on difference quotient arguments developed by the first author [Z. Anal. Angew. 15, 637-650 (1996; Zbl 0858.35069)] for elliptic $$p$$-Laplace type equations.

##### MSC:
 35K65 Degenerate parabolic equations 35K15 Initial value problems for second-order parabolic equations
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##### References:
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