On a comparison principle for Trudinger’s equation

Abstract

We study the comparison principle for non-negative solutions of the equation\frac{\partial(|v|^{p-2}v)}{\partial t}=\operatorname{div}(|\nabla v|^{p-2}% \nabla v),\quad 1<p<\infty.This equation is related to extremals of Poincaré inequalities in Sobolev spaces. We apply our result to obtain pointwise control of the large time behavior of solutions.

Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights

“…The equation in (4.1) covers many of the most widely used mathematical and physical models, such as the porous medium equation (p = 2, 0 < q < 1), the parabolic p-Laplace equation (q = 1) and Trudinger's equation (q = p−1). It is worth mentioning that Trudinger's equation was originally observed in [72] and it has an interesting link to extremals of Poincaré inequalities as shown in [58]. With regard to the Hölder continuity of solutions to problem (4.1) in the unweighted case, the theory of p = 2 is relatively complete, while the wellstudied ranges in the case of p = 2 mainly concentrate in the following two cases: the first is q > p − 1, 1 < p < 2; the second is q ≤ p − 1, p > 2.…”

Interior regularity to the signed solution of the singular doubly nonlinear parabolic equations

“…It also has connections to physical models, such as the dynamics of glaciers in (Mahaffy, 1976), shallow water flows in (Alanso et al, 2008;Feng and Molz, 1997;Hromadka et al, 1985), and frictiondominated flow in a gas network in ( Leugering and Mophou, 2018). Another natural connection between the Trudinger equation and the non-linear eigenvalue issue (Lindgren and Lindqvist, 2022) is that it is crucial to nonlinear potential theory.V. B. Ogelein, F. Duzzar, and N. Liao examined the Hölder continuity of signed solutions for broader equations under structural constraints in (Bogelion et al, 2021).…”

Gradient Hölder regularity for degenerate parabolic systems

“…Proposition 4.14), the same result remains elusive for doubly non-linear equations. See a recent contribution [59] in this connection.…”