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On non-uniformly parabolic functional differential equations. (English) Zbl 1174.35054
In this paper the authors consider weak solutions of quasilinear equations of the form \[ D_t u - \sum_{i=1}^{n}D_i[a_i(t,x,u(t,x),Du(t,x);u)]+a_0(t,x,u(t,x),Du(t,x);u)=f \] in \(Q_T=(0,T) \times \Omega\) (where \(\Omega \in \mathbb R^n\)) with certain homogeneous boundary and initial condition. The given functions \(a_i\) satisfy conditions which are generalizations of the usual conditions for quasilinear parabolic differential equations used in the theory of monotone operators. However, the equation is not assumed to be uniformly parabolic. In the paper the existence of the weak solution is proven under given set of conditions, for any right hand side in the equation. The boundedness and the stabilization of the solution is also investigated.

35K15 Initial value problems for second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
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