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Strong solutions in $$L^ 1$$ of degenerate parabolic equations. (English) Zbl 0828.35050
The paper deals with the degenerate parabolic problem of the form $$u_t = \Delta \varphi (u) + \text{div} F(u)$$ on $$Q = (0,T) \times \Omega$$, $$u(0, \cdot) = u_0$$ on $$\Omega$$, where $$\Omega \subset \mathbb{R}^N$$, $$\varphi \in C^1 (\mathbb{R})$$, $$F \in C^1 (\mathbb{R})^N$$, $$u_0 \in L^\infty (\mathbb{R}^N)$$. Assuming that $$\varphi' > 0$$ a.e. on $$\mathbb{R}$$ and $$|F' |^2 \leq \sigma \varphi'$$ for some $$\sigma \in C (\mathbb{R})$$, the authors prove that the above problem has a strong solution $$u$$, i.e. $$u_t$$, $$\Delta \varphi (u)$$, $$\text{div} F(u)$$ are functions in $$L^1_{\text{loc}} (Q)$$. The proof is based on the theory of $$BV$$ functions in several variables and geometric measures.
Reviewer: O.Titow (Berlin)

##### MSC:
 35K10 Second-order parabolic equations 35K65 Degenerate parabolic equations
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