## 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

### MSC:

 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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### References:

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