Boundary behavior and monotonicity estimates for solutions to nonlinear diffusion equations
The paper deals with the parabolic equation \[ \nu_ t=A(\nu)\nu_{xx}\quad on\quad (0,1)\times {\mathbb{R}}^+,\quad \nu =0\quad on\quad (0,1)\times \{0\}, \] supplemented with boundary conditions on \(\{0\}\times {\mathbb{R}}^+\) and on \(\{1\}\times {\mathbb{R}}^+\) either of Dirichlet (temperature controlled case) or of Neumann type (flux controlled case). This problem is substantially equivalent to the more standard equation in divergence form \(u_ t=(a(u)u_ x)_ x\). After establishing certain compatibility conditions between Dirichlet and Neumann data, the author proves i) a-priori estimates for the solution and its space gradient; ii) monotone dependence of the boundary values on the coefficient A. Specifically, if \[ \partial_ t\nu_ i=A_ i(\nu)\partial_{xx}\nu_ i(x,t),\quad 0<x<1,\quad 0<t<T,\quad \nu_ i=0\quad on\quad (0,1)\times \{0\}, \] \[ -\partial_ x\nu_ i(0,t)=g_ 0(t);\quad \partial_ x\nu_ i(1,t)=0,\quad 0<t<T \] for \(i=1,2\), and the coefficients \(A_ i\) satisfy, beside certain general properties, \(A_ 1(0)=A_ 2(0)\), \(A_ 1'(0)>A_ 2'(0)\), then for \(0<\sigma_ 1<\sigma_ 2\) there exist \(t_ 1=t_ 1(\sigma_ 1)\), \(T_ 1=T_ 1(\sigma_ 2)\) such that \(0<t_ 1<T_ 1\) and \[ (a)\quad \nu_ 1(0,t)-\nu_ 2(0,T)\geq C(t-t_ 1)\lambda \quad for\quad t_ 1\leq t\leq T_ 1, \] where \(C>0\) and \(\lambda =\min_{s\in [\sigma_ 1,\sigma_ 2]}(A_ 1(s)-A_ 2(s));\) moreover, (b) for \(\sigma >0\) there exists \(\tau =\tau (\sigma)>0\) such that \[ \nu_ 1(0,t)-\nu_ 2(0,t)\leq K(\tau)\mu \quad for\quad 0\leq t\leq \sigma, \] where \(K(\tau)>0\), and \(\mu =\max_{s\in [0,\sigma]}(A_ 1(s)-A_ 2(s))\). a) shows that identical flux controlled experiments involving distinct coefficients \(A_ 1\), \(A_ 2\) in the appropriate class cannot produce identical Dirichlet data at \(x=0\); b) then shows that identical flux controlled experiments with identical coefficients must produce identical Dirichlet boundary values at \(x=0\). Similar results prevail in the temperature controlled case. These properties, which are proved in a careful and detailed way, are relevant when analysing inverse problems for nonlinear parabolic equations.
scientific article; zbMATH DE number 3955553