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A boundary value problem for a class of quasilinear operators of Fokker-Planck type. (English) Zbl 0881.35025

Summary: We study a boundary value problem for a class of nonlinear operators of the following type: \[ Lu:= \sum^{p_0}_{i,j=1} a_{ij}(z,u)\partial_{ij}u+\langle x,BD_xu\rangle-\partial_tu, \] where \(B\) is a suitable \(N\times N\) constant matrix, \(A_0=(a_{ij}(z,u))\) is a \(p_0\times p_0\) matrix which is positive definite in \(\mathbb{R}^{p_0}\), uniformly with respect to \((z,u)\in\Omega\times\mathbb{R}\). Here \(\Omega\) is a bounded open subset of \(\mathbb{R}^{N+1}\) and \(z=(x,t)\) denotes the point in \(\mathbb{R}^N\times \mathbb{R}\).

MSC:

35G30 Boundary value problems for nonlinear higher-order PDEs
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
65H10 Numerical computation of solutions to systems of equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B45 A priori estimates in context of PDEs
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