Asympotic behaviors of spatial critical points and zeros of solutions of parabolic equations

(1)Level surfaces invariant with time of solutions of diffusion equationsWe consider solutions of the initial-Neumann problem for the heat@equation on bounded Lipschitz domains in Euclidean space，and with the help of the classification theorem of isoparametric hypersurfaces in Euclidean space of Levi-Civita(1937)and Segre(1938)，we classify the solutions whose isothermal surfaces are invariant with time with time。Furthermore，we can deal with nonlinear diffusion equations such as the porous medium@equation，and we get similar classification theorems.(2)Asymptotic behaviors of the interfaces with sign changes of solutions of the one-dimensional porous medium equationWe consider the Cauchy and the initial-Dirichlet problems for the one-dimensional evolution p-Laplacian@equation with p&gt；1for nonzero，bounded and nonneginitial data compavort。It was shown that after a finite time the set of spatial critical points of the solution u in{u&gt；0}consists of one point，say x=…More x(T)for time t.In this research，we show that after a finite time x(T)is C I D11文件D1 in t.Furthermore，we can deal with generalized porous medium equations with sign changes，and we get C I D 11文件D1 regularity of the interfaces with sign changes.Also，in the initial-Dirichlet problem for the one-dimensional evolution p-Laplacian@equation，we show that there exists a positive constantβ=β(ρ)such that x(T)t i D1-βii D1 tends to some positive constant as t→∞.(3)Stationary critical points of the heat flow and the symmetries of the domainsWe consider the initial-Dirichlet problem for the heat@equation on bounded and simply connected domains in the plane。By a new method with the help of the Riemann Mapping theorem in complex analysis，we give a characterization of domains invariant under the rotation of angle2π/3 by making use of the stationary critical points of the heat flow。(Previously，only the characterizations of balls and centrosymmetric domains were obtained.)Furthermore，we consider stationary critical points of the heat flow in sphere S I D1N文件D1 and in hyperbolic space H文件D1N文件D1，and prove several results corresponding to those in Euclidean space which have been proved in Magnanini and Sakaguchi(1997，1999)。Precisely.We get the characterizations of geodesic balls and centrosymmetric domains by making use of the stationary critical points of the heat flow。Less：Less

(1) Level surfaces invariant with time of solutions of diffusion equationsWe consider solutions of the initial-Neumann problem for the heat equation on bounded Lipschitz domains in Euclidean space, and with the help of the classification theorem of isoparametric hypersurfaces in Euclidean space of Levi-Civita (1937) and Segre (1938), we classify the solutions whose isothermal surfaces are invariant with time. Furthermore, we can deal with nonlinear diffusion equations such as the porous medium equation, and we get similar classification theorems.(2) Asymptotic behaviors of the interfaces with sign changes of solutions of the one-dimensional porous medium equationWe consider the Cauchy and the initial-Dirichlet problems for the one-dimensional evolution p-Laplacian equation with p>1 for nonzero, bounded, and nonnegative initial data having compact support. It was shown that after a finite time the set of spatial critical points of the solution u in {u > 0} consists of one point, say x = … More x(t) for time t. In this research, we show that after a finite time x(t) is CィイD11ィエD1 in t. Furthermore, we can deal with generalized porous medium equations with sign changes, and we get CィイD11ィエD1 regularity of the interfaces with sign changes. Also, in the initial-Dirichlet problem for the one-dimensional evolution p-Laplacian equation, we show that there exists a positive constant β=β(ρ) such that x(t)tィイD1-βィエD1 tends to some positive constant as t → ∞.(3) Stationary critical points of the heat flow and the symmetries of the domainsWe consider the initial-Dirichlet problem for the heat equation on bounded and simply connected domains in the plane. By a new method with the help of the Riemann Mapping theorem in complex analysis, we give a characterization of domains invariant under the rotation of angle 2π/3 by making use of the stationary critical points of the heat flow. (Previously, only the characterizations of balls and centrosymmetric domains were obtained.) Furthermore, we consider stationary critical points of the heat flow in sphere SィイD1NィエD1 and in hyperbolic space HィイD1NィエD1, and prove several results corresponding to those in Euclidean space which have been proved in Magnanini and Sakaguchi (1997, 1999). Precisely. We get the characterizations of geodesic balls and centrosymmetric domains by making use of the stationary critical points of the heat flow. Less