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

Abstract

We consider the Cauchy-Neumann problem of the heat equation in the exterior domain of a ball in R N , and study the movement of hot spots of the solution as t → ∞.

Asymptotic analysis of the p-Laplacian flow in an exterior domain

“…Application to the linear case, p = 2. In the linear case p = 2, a partial inner behaviour in the exterior of a ball is described in [14]. In this particular case, our results coincide with those of that paper in N 3.…”

“…In the case of the linear heat equation the analysis is made easier by the possibility of using integral representation of the solutions, cf. Ishige [13] and [14]. In the case of the porous medium equation, the asymptotic behaviour in the whole space is well known cf.…”

“…In this particular case, our results coincide with those of that paper in N 3. Since [14] treats only asymptotic behaviour in fixed compact sets, in dimension N = 2 a logarithmic improvement of the decay rate is proved. This improvement follows also easily in our case in a fixed compact set, but not in the general inner behaviour.…”

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

“…In this paper we extend the arguments in [6][7][8], and study the large time behavior of the hot spots H (t) of the solution u of (1.1) under the condition (V ). In particular, under the condition (V ), the hot spots H (t) run away to the space infinity as t → ∞, and we study the rate and the direction for the hot spots to run away as t → ∞.…”

“…Furthermore they also treated the movement of hot spots for the radial solutions in the exterior domain of a ball. Subsequently, the first author of this paper [6,7] studied the movement of hot spots on the exterior domain of a ball without the radial symmetry of the initial data, by using rescale arguments and the radial symmetry of the domain effectively. Recently the authors of this paper [8] studied the decay rate of derivatives of the solution of the heat equations with a potential, and proved that the optimal decay rate of the derivatives of solutions was determined by the shape of the harmonic functions for − V .…”