Movement of Hot Spots over Unbounded Domains in RN

Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition

“…As stated in [6], it is difficult to know the sign of differential of the Neumann heat kernel even for the case that is the exterior of a ball, and so it seems difficult to obtain Theorems 1.1 and 1.2 by using the fundamental properties of the Neumann heat kernel. In order to prove Theorems 1.1 and 1.2, we consider the asymptotic behavior of the radial solution v k of the Cauchy-Neumann problem (L k ):…”

“…In particular, for the Euclidean space R N , they proved that, for any nonzero, nonnegative initial data ∈ L ∞ c (R N ), the hot spots H (t) of the solution at each time t > 0 are contained in the closed convex hull of the support of , and H (t) tends to the center of mass of as t → ∞. Subsequently, Jimbo and Sakaguchi [6] studied the movement of hot spots of the solution of the heat equation in the half space R N + and in the exterior domain of a ball, under boundary conditions. In particular, they proved that the hot spots H (t) of the solution of (1.1) in the half space R N + with the nonzero, nonnegative initial data ∈ L ∞ c (R N + ) satisfies…”

Asymptotic Behavior of Zeros of Solutions for Parabolic Equations

“…Here the sets H(t) and C(t) of hot spots and of spatially critical points of a solution u are defined as Their results told that there are T>0 and x(t) # R N such that C(t)= H(t)=[x(t)] for t T and x(t) converges to the barycenter of initial data. In [8], they also studied the case of the half space or exterior domain with various boundary conditions. However it seems that there is no result on the asymptotic behavior of zeros of sign-changing solutions more generally.…”

“…In the case of :=0, the movement of hot spots and spatially critical points of positive solutions with initial data having compact support was investigated in detail in [2,8,13]. Here the sets H(t) and C(t) of hot spots and of spatially critical points of a solution u are defined as Their results told that there are T>0 and x(t) # R N such that C(t)= H(t)=[x(t)] for t T and x(t) converges to the barycenter of initial data.…”

Large time behaviors of hot spots for the heat equation with a potential

“…In particular, they used the fundamental solution of the heat equation on R N , and proved that the hot spots on R N tend to the center of mass of the initial data as t → ∞ if the initial data is a nonnegative function having a compact support. Jimbo and Sakaguchi [9] treated the heat equation on the half-space in R N , and studied the relation between the movement of hot spots and the boundary conditions. Furthermore they also treated the movement of hot spots for the radial solutions in the exterior domain of a ball.…”