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On the regularity of generalized solutions of quasilinear degenerate parabolic systems of second order. (English. Russian original) Zbl 0748.35008

Proc. Steklov Inst. Math. 188, 25-63 (1991); translation from Tr. Mat. Inst. Steklova 188, 22-52 (1990).
The author proves the boundedness and Hölder continuity of generalized solutions for second order quasilinear degenerate parabolic systems of the form \[ \partial u^ i/\partial t-\partial/\partial x_ \alpha(| u|^{2\sigma} a^{\alpha\beta}(x,t,u,\nabla u)\partial u^ i/\partial x_ \beta)+b_ i(x,t,u,\nabla u)=0, \] where \(\sigma>0\), \(\gamma|\xi|^ 2\leq a^{\alpha\beta}(x,t,u,\nabla u)\xi_ \alpha \xi_ \beta\leq\mu|\xi|^ 2\), \(\nu,\mu>0\), \(x=(x_ 1,\ldots,x_ n)\), \(u=(u^ 1,\ldots,u^ N)\), \(\nabla u=(u^ i_ \alpha)_{i=1,\ldots,N; \alpha=1,\ldots,n}\), \(u^ i_ \alpha=\partial u^ i/\partial x_ \alpha\), \((x,t)\in Q_ T=\Omega\times(0,T)\), \(\Omega\subset R^ n\), \(n\geq 1\), \(N\geq 1\), \(T>0\).

MSC:

35D10 Regularity of generalized solutions of PDE (MSC2000)
35K65 Degenerate parabolic equations
35B45 A priori estimates in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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