Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

To the uniqueness problem for nonlinear parabolic equations

Pages: 315 - 336, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.315

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H. Gajewski - WeierstraƟ Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117, Berlin, Germany (email)
I. V. Skrypnik - Institute for Applied Mathematics and Mechanics, Rosa Luxemburg Str. 74, 340114, Donetsk, Ukraine (email)

Abstract: We prove a priori estimates in $L^2(0,T;W^{1,2}(\Omega)) \cap L^\infty(Q)$, existence and uniqueness of solutions to Cauchy--Dirichlet problems for parabolic equations

$\frac {\partial \sigma(u)}{\partial t} - \sum_{i=1}^n \frac {\partial}{\partial x_i}${$\rho(u) b_i (t,x,\frac{\partial u}{\partial x})$} $ + a (t,x,u,\frac{\partial u}{\partial x}) = 0,$

$(t,x) \in Q = (0,T) \times \Omega$, where $\rho(u) = \frac{d }{du}\sigma(u)$. We consider solutions $u$ such that $\rho^{\frac{1}{2}}(u) | \frac{\partial u}{\partial x} | \in L^2 (0,T;L^2 (\Omega ) ), \frac {\partial }{\partial t}\sigma(u) \in L^2 ( 0,T;[ W^{1,2} ( \Omega ) ]^\star ). $
Our nonstandard assumption is that log$\rho (u)$ is concave. Such assumption is natural in view of drift diffusion processes for example in semiconductors and binary alloys, where $u$ has to be interpreted as chemical potential and $\sigma$ is a distribution function like $\sigma=e^u$ or $\sigma=\frac {1}{1+e^{-u}}$.

Keywords:  Nonlinear parabolic equations, bounded solutions, uniqueness, nonstandard assumptions, degenerate typ.
Mathematics Subject Classification:  35B45, 35K15, 35K20 35K65.

Received: May 2001;      Revised: September 2002;      Available Online: October 2003.