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A new regularity estimate for solutions of singular parabolic equations

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  • In 1982, K. Ecker showed that solutions of the parabolic equation \[ u_t=\operatorname {div} \left( \frac {Du}{(1+|Du|^2)^{1/2}}\right) + H(x,u) \] have a very unusual regularity property: The interior regularity of $u$ is determined only by its initial regularity. In this note, we show that a similar result is true for a general class of equations. The model equation is \[ u_t=\operatorname {div} \left( |Du|^{p-2} {Du}\right) \] with $1

    Mathematics Subject Classification: Primary: 35K15, 35B65; Secondary: 35K55, 35B45.

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