WIAS Preprint No. 658, (2001)

To the uniqueness problem for nonlinear parabolic equations



Authors

  • Gajewski, Herbert
  • Skrypnik, Igor V.

2010 Mathematics Subject Classification

  • 35B45 35K15 35K20 35K65

Keywords

  • Nonlinear parabolic equations, bounded solutions, uniqueness, nonstandard assumptions, degenerate type

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 $$ fracpartial sigma(u)partial t - sum_i=1^n fracpartialpartial x_i Big rho(u) b_i Big (t,x,fracpartial upartial x Big ) Big + a Big (t,x,u,fracpartial upartial x Big ) = 0, $$ $(t,x) in Q = (0,T) times Omega$, where $rho(u) = fracddusigma(u)$. We consider solutions $u$ such that $rho^frac12(u) left fracpartial upartial x right in L^2 ( 0,T,L^2 (Omega)), fracpartialpartial tsigma(u) in L^2 (0,T, [ cirWhspace*-0.1mm^1,2 (Omega) ]^ast )$. 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 $sigma=frac11+e^u$.

Appeared in

  • Discrete Contin. Dyn. Syst., 9 (2003)

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