To the uniqueness problem for nonlinear parabolic equations doi:10.3934/dcds.2004.10.315 Abstract Full Text (254.1K) Related Articles
H. Gajewski  Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D10117, Berlin, Germany (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 CauchyDirichlet 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 ). $
Keywords: Nonlinear parabolic equations, bounded solutions, uniqueness, nonstandard assumptions, degenerate typ.
Received: May 2001; Revised: September 2002; Published: October 2003. 
2013 IF (1 year).923
