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The Cauchy problem for a quasilinear parabolic equation with gradient absorption. (English. Russian original) Zbl 1252.35160

Sb. Math. 203, No. 4, 581-611 (2012); translation from Mat. Sb. 203, No. 4, 131-160 (2012).
Summary: The qualitative properties of solutions to the Cauchy problem for a degenerate parabolic equation containing a nonlinear operator of Baouendi-Grushin type and with gradient absorption whose density depends on time as well as space variables, are investigated. Bounds for the diameter of the support of the solution which are sharp with respect to time are obtained, together with its maximum. A condition is discovered which determines whether or not the phenomenon of decay to zero of the total mass of the solution occurs.

MSC:

35K59 Quasilinear parabolic equations
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
35K15 Initial value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
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