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On parabolic initial-boundary value problems with critical growth for the gradient. (English) Zbl 0836.35078
Summary: We prove the existence of weak solutions for the initial-boundary value problem of the quasilinear parabolic equation \[ {\partial u \over \partial t} + A(u) + g(x,t,u,Du) = f. \] Here \(A\) is a Leray-Lions type operator from \({\mathcal V} = L^p (0,T; W_0^{1,p} (\Omega)) \) to its dual space \({\mathcal V}^*\), \(g\) is a nonlinear term with critical growth with respect to \(Du\) satisfying a sign condition and no growth condition with respect to \(u\); \(f\) is a given element in \({\mathcal V}^*\).

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
47H05 Monotone operators and generalizations
35K20 Initial-boundary value problems for second-order parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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