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Quasilinear parabolic equations containing a Volterra operator in the coefficients. (English. Russian original) Zbl 0683.35042
Math. USSR, Sb. 64, No. 2, 527-542 (1989); translation from Mat. Sb., Nov. Ser. 136(178), No. 4(8), 530-545 (1988).
Consider the first initial-boundary value problem for the quasilinear parabolic equations \[ u_ t+(-1)^ m\sum_{| \alpha | =m}D^{\alpha}[a_{\alpha}(\int^{t}_{0}| D^{\alpha}u|^ qdt)| D^{\alpha}u|^{q-2}D^{\alpha}u]=f \] and \[ u_ t+(- 1)^ m\sum_{| \alpha | =m}D^{\alpha}[a(\int^{t}_{0}| D^ mu|^ qdt)| D^ mu|^{q-2}D^{\alpha}u]=f. \] Conditions of the global solvability of the problems are established.
Reviewer: J.Tian

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K30 Initial value problems for higher-order parabolic equations
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