Singular perturbations for parabolic equations with unbounded coefficients leading to ultraparabolic equations. (English) Zbl 1164.35312

Akhmetov, Denis R.; Lavrentiev, Mikhail M. jun.; Spigler, Renato

Differ. Integral Equ. 17, No. 1-2, 99-118 (2004).

Summary: Linear parabolic equations with coefficients of the lower-order terms unbounded and with a small parameter multiplying some of the second (highest) space derivatives are considered, in the limiting case when such a parameter goes to zero. This yields a degenerate parabolic (ultraparabolic) equation with one space-like variable, \(x\), and two time-like variables, \(y\) and \(t\). No boundary-layer is found to be needed in the case of the boundary-value problem on the \(x\)-unbounded domain \(Q_T=\{(x,y,t)\in \mathbb R\times [0,1]\times [0,T]\}\) with a periodic boundary condition in the variable \(y\) and initial data at \(t=0\).

MSC:

35B25	Singular perturbations in context of PDEs
35F15	Boundary value problems for linear first-order PDEs
35F30	Boundary value problems for nonlinear first-order PDEs
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

degenerate parabolic equation; ultraparabolic equation; small parameter