Interaction between nonlinear diffusion and geometry of domain

Abstract

Let Ω be a domain in R N , where N ≥ 2 and ∂Ω is not necessarily bounded. We consider nonlinear diffusion equations of the form ∂ t u = ∆φ(u). Let u = u(x, t) be the solution of either the initial-boundary value problem over Ω, where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set R N \ Ω.We consider an open ball B in Ω whose closure intersects ∂Ω only at one point, and we derive asymptotic estimates for the content of substance in B for short times in terms of geometry of Ω. Also, we obtain a characterization of the hyperplane involving a stationary level surface of u by using the sliding method due to Berestycki, Caffarelli, and Nirenberg. These results tell us about interactions between nonlinear diffusion and geometry of domain.Key words. nonlinear diffusion, geometry of domain, initial-boundary value problem, Cauchy problem, initial behavior.

Interaction between fast diffusion and geometry of domain

“…When p > 2, m > 1, α = 1, and f ≡ g ≡ 1, the same formulas (1.7) and (1.12) were obtained for problems (1.3)-(1.5) and (1.8)-(1.10) in [MS1]. With the aid of the techniques employed in [MS3], one can easily see that the formulas (1.7) and (1.12) also hold true for problems (1.6) and (1.11). Moreover, in [MS3], the nonlinear diffusion equation of the form ∂ t u = ∆φ(u) where δ 1 ≤ φ ′ (s) ≤ δ 2 (s ∈ R) for some positive constants δ 1 and δ 2 was also dealt with.…”

“…With the aid of the techniques employed in [MS3], one can easily see that the formulas (1.7) and (1.12) also hold true for problems (1.6) and (1.11). Moreover, in [MS3], the nonlinear diffusion equation of the form ∂ t u = ∆φ(u) where δ 1 ≤ φ ′ (s) ≤ δ 2 (s ∈ R) for some positive constants δ 1 and δ 2 was also dealt with. By a little more observation, we see that any α > 0 is OK for these cases.…”

“…This property does not occur when 1 < p < 2 or 0 < m < 1, because of the property of infinite speed of propagation of disturbances from rest. Also in [MS3], the equation ∂ t u = ∆φ(u) has the property of infinite speed of propagation of disturbances from rest.…”

The Matzoh Ball Soup Problem: A complete characterization

“…In a subsequent series of papers, the same authors extended their result in several directions: spherical symmetry also holds for certain evolution nonlinear equations [11,13,15,16]; a hyperplane can be characterized as an invariant equipotential surface in the case of an unbounded solid that satisfies suitable sufficient conditions [12,14]; for a certain Cauchy problem, a helicoid is a possible invariant equipotential surface [9]; spheres, infinite cylinders and planes are characterized as (single) invariant equipotential surfaces in R 3 [8]; similar symmetry results can also be proven in the sphere and the hyperbolic space [13].…”

Asymptotics for the resolvent equation associated to the game-theoreticp-laplacian

“…uniformly on a bounded domain Ω, where d Γ (x) denotes the distance of a point x ∈ Ω to Γ. Further evidence of the influence of geometry on the asymptotic behavior of solutions of elliptic and parabolic equations for small values of the relevant parameter was given by the second author of this note and S. Sakaguchi in a series of papers both in the linear case ([MS1], [MS3], [MS5], [MS7], [MM]) and in certain non-linear contexts ([MS2], [MS4], [MS6], [Sa1]), concerning both initial-boundary value problems ( [MS1], [MS2], [MS4]) and initial-value problems ( [MPeS], [MPrS]), and even for two-phase problems ([Sa2], [Sa3], [CMS]). For instance, in [MS1,Theorem 2.3], in the case p = 2 for problem (1.1)-(1.2), the following formula involving the mean value of u ε was established:…”