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We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation ∂ t u ε , δ + div f ε , δ ( x , u ε , δ ) = ε Δ u ε , δ + δ ( ε ) ∂ t Δ u ε , δ , x ∈ M , t ≥ 0 u | t = 0 = u 0 ( x ) . Here, f ε , δ and u 0 are smooth functions while ε and δ = δ ( ε ) are fixed constants. Assuming f ε , δ → f ∈ L p ( R d × R ; R d ) for some 1 less then p less then ∞ , strongly as ε → 0 , we prove that, under an appropriate relationship between ε and δ ( ε ) depending on the regularity of the flux
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