We consider semilinear equations of the form p(D)u=F(u), with a locallybounded nonlinearity F(u), and a linear part p(D) given by a Fouriermultiplier. The multiplier p(\xi) is the sum of positively homogeneous terms,with at least one of them non smooth. This general class of equations includesmost physical models for traveling waves in hydrodynamics, the Benjamin-Onoequation being a basic example. We prove sharp pointwise decay estimates forthe solutions to such equations, depending on the degree of the non smoothterms in p(\xi). When the nonlinearity is smooth we prove similar estimates forthe derivatives of the solution, as well holomorphic extension to a strip, foranalytic nonlinearity.
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