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An \(L^p\)-approach for the study of degenerate parabolic equations. (English) Zbl 1073.35099

Summary: We give regularity results for solutions of a parabolic equation in non-rectangular domains \(U=\bigcup_{t\in ] 0,1[}\{ t\} \times I_{t}\) with \(I_{t}=\{x:0<x<\varphi (t)\}\). The optimal regularity is obtained in the framework of the space \(L^{p}\) with \(p>3/2\) by considering the following cases: (1) When \(\varphi (t)=t^{\alpha }\), \(\alpha> 1/2\) with a regular right-hand side belonging to a subspace of \(L^{p}(U)\) and under assumption \(p>1+\alpha \). (2) When \(\varphi (t)=t^{1/2}\) with a right-hand side taken only in \(L^{p}(U)\).

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
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