UNIFORM BOUNDEDNESS FOR WEAK SOLUTIONS OF QUASILINEAR PARABOLIC EQUATIONS

In this paper, we study the boundedness of weak solutions to quasilinear parabolic equations of the form u(t) - div A(x, t, del u) = 0, where the nonlinearity A(x, t, del u) is modelled after the well-studied p-Laplace operator. The question of boundedness has received a lot of attention over the past several decades with the existing literature showing that weak solutions in either 2N/N + 2 < p < 2, p = 2, or 2 < p, are bounded. The proof is essentially split into three cases mainly because the estimates that have been obtained in the past always included an exponent of the form 1/p-2 or 1/2-p, which blows up as p -> 2. In this note, we prove the boundedness of weak solutions in the full range 2N/ N + 2 < p < infinity without having to consider the singular and degenerate cases separately. Subsequently, in a slightly smaller regime of 2N/N + 1 < p < infinity, we also prove an improved boundedness estimate.