Nodal properties of solutions of parabolic equations. (English) Zbl 0727.35005
We review some known facts about the zero set of a solution of a scalar parabolic equation \(u_ t=a(x,t)u_{xx}+b(x,t)u_ x+c(x,t)u\), \(x_ 0<x<x_ 1\), \(0<t<T.\)
In particular, we discuss some applications to spectral theory, the dynamics of nonlinear diffusion equations, and the geometric heat equation for plane curves.
In particular, we discuss some applications to spectral theory, the dynamics of nonlinear diffusion equations, and the geometric heat equation for plane curves.
MSC:
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35K05 | Heat equation |
| 35K57 | Reaction-diffusion equations |
| 35P99 | Spectral theory and eigenvalue problems for partial differential equations |
Keywords:
nodal set; scalar parabolic equation; spectral theory; nonlinear diffusion equations; geometric heat equationReferences:
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