$L_{{\text{loc}}}^\infty $-Estimates for Local Solutions of Degenerate Parabolic Equations
Daniele Andreucci · SIAM Journal on Mathematical Analysis · 1991
A local sup-estimate for local (sub)solutions of degenerate parabolic equations is proved. Such local estimates do not involve any dependence on the initial and boundary data, but rather, provide a bound for the solution in a given domain, only in terms of some integral norm of the solution in a larger domain. Estimates of this kind for solutions of linear nondegenerate parabolic equations are due to Moser (Comm. Pure Appl. Math., 17 (1964), pp. 101–134). Sharp estimates of this kind have not been available in the literature in the degenerate case. The new input here is an interpolation process that permits one to deal with the degeneracy of the equation. The estimates shown here are sharp and reduce to the classical ones of Moser in the linear nondegenerate case.