Abstract
Abstract We study the comparison principle for non-negative solutions of the equation ∂ ( | v | p - 2 v ) ∂ t = div ( | ∇ v | p - 2 ∇ v ) , 1 < p < ∞ . \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.
Highlights
Among the so-called doubly non-linear evolutionary equations, Trudinger’s equation 1 < p < ∞, (1.1)is distinguished
The equation was originally considered by Trudinger in [22], where a Harnack inequality was studied for a wider class of evolutionary equations
The equation has two special features: it is homogeneous and it is not translation invariant, except for p = 2 when it reduces to the heat equation
Summary
Among the so-called doubly non-linear evolutionary equations, Trudinger’s equation. is distinguished. We shall study its solutions in ΩT = Ω × (0, T), where Ω is a domain in Rn. The equation was originally considered by Trudinger in [22], where a Harnack inequality was studied for a wider class of evolutionary equations. Our result implies that for p ≥ 2, all non-negative continuous weak solutions are viscosity solutions in the sense of Crandall, Evans, and Lions in [3] Trudinger’s equation has an interesting connection to extremals of Poincaré inequalities in the Sobolev space W1,p(Ω). In [8], a comparison principle is proved for a general class of doubly non-linear equations. In the Appendix, we give a proof of the fact that the maximum of a subsolution and a constant is again a subsolution
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