Uniqueness for a forward backward diffusion equation with smooth constitutive funcation

Abstract

We show that the initial boundary value problem for the diffusion equation u_L ⊘(u_x)_x with bounded Neumann boundary conditions has at most one smooth solutions on the infinite cylinder provided the nonmonotonic function ⊘ is piecewise continuously differentiable on the reals has at most finitely many extreme values on any bounded interval,satisfies the coercivity condition s⊘(s) ≤ cs², c < 0 and has a derivative which is nonzero almost everywhere. In particular this result provides uniquencess for the model cubic ⊘(s)=s³ /3-3s²/2+2s proposed by Hölling and Nohel

AMS(MOS)::

• 35K20
• 35K55

Keywords:

• Uniqueness
• diffusion
• initial boundary value problem
• non linear
• nonmonotone
• constitutive function
• Neumann boundary conditions