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