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A counterexample in regularity theory for parabolic systems. (English) Zbl 0573.35053
The authors study the degenerate parabolic partial differential equation \[ -(\partial /\partial x_ i)(a_{ij}(x,t)(\partial u/\partial x_ j))+(\partial u/\partial t)=f \] under suitable assumptions on the coefficients, deriving global properties of the solutions, including the \(L^ 2\) continuity of solutions to the Cauchy-Dirichlet problem. They construct examples to show that, when \(f=0\), the equation fails to have local \(L^{\infty}\) estimates unlike the elliptic case. Other examples are provided to establish that Harnack’s inequality cannot hold.
Reviewer: O.T.Haimo

35K65 Degenerate parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
35K20 Initial-boundary value problems for second-order parabolic equations
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