Boundary estimates for non-negative solutions to non-linear parabolic equations

Abstract

Boundary estimates for non-negative solutions to non-linear parabolic equations.Access to the published version may require subscription. N.B. When citing this work, cite the original published paper. Abstract This paper is mainly devoted to the boundary behavior of non-negative solutions to the equationwhere Ω ⊂ R n is a bounded non-tangentially accessible (NTA) domain and T > 0. The assumptions we impose on A imply that H is a non-linear parabolic operator with linear growth. Our main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary. Furthermore, to each such solution one can associate a natural Riesz measure supported on the lateral boundary and one of our main result is a proof of the doubling property for this measure. Our results generalize, to the setting of non-linear equations with linear growth, previous results concerning the boundary behaviour, in Lipschitz cylinders and time-independent NTA-cylinders, established for non-negative solutions to equations of the type ∂ t u − ∇ · (A(x, t)∇u) = 0, where A is a measurable, bounded and uniformly positive definite matrixvalued function. In the latter case the measure referred to above is essentially the caloric or parabolic measure associated to the operator and related to Green's function. At the end of the paper we also remark that our arguments are general enough to allow us to generalize parts of our results to general fully non-linear parabolic partial differential equations of second order.

Boundary behavior of solutions to the parabolic p-Laplace equation

“…On the contrary for p ≠ 2, 1 < p < ∞, much less is known concerning these problems and we refer the reader to [2,3,24] for accounts of the current literature. For a relatively complete picture in the case of non-linear parabolic operators with linear growth we refer to [30]. However, it is also important to mention that there is an interesting and related recent literature devoted to the asymptotic and pointwise behavior of solutions to non-linear diffusion equations on bounded domains, see [32], and also [8] for the porous medium type equations, and the references therein.…”

Boundary behavior of solutions to the parabolic p-Laplace equation

“…On the contrary for p ≠ 2, 1 < p < ∞, much less is known concerning these problems and we refer the reader to [2,3,24] for accounts of the current literature. For a relatively complete picture in the case of non-linear parabolic operators with linear growth we refer to [30]. However, it is also important to mention that there is an interesting and related recent literature devoted to the asymptotic and pointwise behavior of solutions to non-linear diffusion equations on bounded domains, see [32], and also [8] for the porous medium type equations, and the references therein.…”