Decay rates of the derivatives of the solutions of the heat equations in the exterior domain of a ball

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

“…By [8], we see that U μ k,L is a positive, increasing function in (L, ∞). Furthermore we see that U μ k,L (r) r α k −1 and U μ k,L (r) r α k for all r L + 1.…”

“…However, it also seems difficult to apply the same arguments as in [6][7][8] directly, because of the singularities of the potential (ω + ω k )r −2 at r = 0 and of e s V (e −s/2 y) at (y, s) = (0, ∞). We use several properties of U k and radial functions constructed from inhomogeneous elliptic problems, and construct two super-solutions of (P k ) to overcome the difficulty driven from the singularity of (ω + ω k )r −2 at r = 0.…”

“…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 → ∞.…”

“…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 .…”

“…We follow the strategy in [6][7][8], and obtain the asymptotic behaviors of the derivatives of w k,i (s) as s → ∞. However, it also seems difficult to apply the same arguments as in [6][7][8] directly, because of the singularities of the potential (ω + ω k )r −2 at r = 0 and of e s V (e −s/2 y) at (y, s) = (0, ∞).…”