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Decay to rarefactions for scalar multidimensional viscous conservation laws

We show that solutions to the viscous conservation laws in several space dimensions converge to rarefactions as time tends to infinity, when the initial data belongs to some large global class of functions. The convergence takes place in L ∞ (R n ), and in L P (R×K), where K is compact in R n−1 , and we determine the asymptotic rate