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On a Parabolic Strongly Nonlinear Problem on Manifolds

In this work we will prove the existence uniqueness and asymptotic behavior of weak solutions for the system (*) involving the pseudo Laplacian operator and the condition ∂u ∂t + n ∑ i=1 ∣∣ ∂u ∂xi ∣∣p−2 ∂u ∂xi νi + |u|u = f on Σ1, where Σ1 is part of the lateral boundary of the cylinder Q = Ω × (0, T ) and f is a given function defined on Σ1.