TY - JOUR

T1 - When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?

AU - Sakaguchi, Shigeru

PY - 1999/1/1

Y1 - 1999/1/1

N2 - We consider solutions of the initial-Neumann problem for the heat equation on bounded Lipschitz domains in ℝN; and classify the solutions whose spatial level surfaces are invariant with respect to the time variable. (Of course, the values of each solution on its spatial level surfaces vary with time.) The prototype of such classification is a result of Alessandrini, which proved a conjecture of Klamkin. He considered the initial-Dirichlet problem for the heat equation on bounded domains and showed that if all the spatial level surfaces of the solution are invariant with respect to the time variable under the homogeneous Dirichlet boundary condition, then either the initial data is an eigenfunction or the domain is a ball and the solution is radially symmetric with respect to the space variable. His proof is restricted to the initial-Dirichlet problem for the heat equation. In the present paper, in order to deal with the initial-Neumann problem, we overcome this obstruction by using the invariance condition of spatial level surfaces more intensively with the help of the classification theorem of isoparametric hypersurfaces in Euclidean space of Levi-Civita and Segre. Furthermore, we can deal with nonlinear diffusion equations, such as the porous medium equation.

AB - We consider solutions of the initial-Neumann problem for the heat equation on bounded Lipschitz domains in ℝN; and classify the solutions whose spatial level surfaces are invariant with respect to the time variable. (Of course, the values of each solution on its spatial level surfaces vary with time.) The prototype of such classification is a result of Alessandrini, which proved a conjecture of Klamkin. He considered the initial-Dirichlet problem for the heat equation on bounded domains and showed that if all the spatial level surfaces of the solution are invariant with respect to the time variable under the homogeneous Dirichlet boundary condition, then either the initial data is an eigenfunction or the domain is a ball and the solution is radially symmetric with respect to the space variable. His proof is restricted to the initial-Dirichlet problem for the heat equation. In the present paper, in order to deal with the initial-Neumann problem, we overcome this obstruction by using the invariance condition of spatial level surfaces more intensively with the help of the classification theorem of isoparametric hypersurfaces in Euclidean space of Levi-Civita and Segre. Furthermore, we can deal with nonlinear diffusion equations, such as the porous medium equation.

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U2 - 10.1007/BF02791135

DO - 10.1007/BF02791135

M3 - Article

AN - SCOPUS:0013210055

VL - 78

SP - 219

EP - 243

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

ER -