# American Institute of Mathematical Sciences

April  2016, 36(4): 1869-1880. doi: 10.3934/dcds.2016.36.1869

## Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations

 1 Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 - Torino, Italy 2 Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received  March 2015 Revised  March 2015 Published  September 2015

We consider semilinear equations of the form $p(D)u=F(u)$, with a locally bounded nonlinearity $F(u)$, and a linear part $p(D)$ given by a Fourier multiplier. The multiplier $p(\xi)$ is the sum of positively homogeneous terms, with at least one of them non smooth. This general class of equations includes most physical models for traveling waves in hydrodynamics, the Benjamin-Ono equation being a basic example.
We prove sharp pointwise decay estimates for the solutions to such equations, depending on the degree of the non smooth terms in $p(\xi)$. When the nonlinearity is smooth we prove similar estimates for the derivatives of the solution, as well as holomorphic extension to a strip, for analytic nonlinearity.
Citation: Marco Cappiello, Fabio Nicola. Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1869-1880. doi: 10.3934/dcds.2016.36.1869
##### References:
 [1] C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation - a nonlinear Neumann problem in the plane,, Acta Math., 167 (1991), 107. doi: 10.1007/BF02392447. Google Scholar [2] T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, J. Fluid Mech., 29 (1967), 559. doi: 10.1017/S002211206700103X. Google Scholar [3] H. A. Biagioni and T. Gramchev, Fractional derivative estimates in Gevrey classes, global regularity and decay for solutions to semilinear equations in $\mathbbR^n$,, J. Differential Equations, 194 (2003), 140. doi: 10.1016/S0022-0396(03)00197-9. Google Scholar [4] J. Bona and Y. Li, Analyticity of solitary-wave solutions of model equations for long waves., SIAM J. Math. Anal., 27 (1996), 725. doi: 10.1137/0527039. Google Scholar [5] J. Bona and Y. Li, Decay and analyticity of solitary waves,, J. Math. Pures Appl., 76 (1997), 377. doi: 10.1016/S0021-7824(97)89957-6. Google Scholar [6] N. Burq and F. Planchon, On well-posedness for the Benjamin-Ono equation,, Math. Ann., 340 (2008), 497. doi: 10.1007/s00208-007-0150-y. Google Scholar [7] M. Cappiello, T. Gramchev and L. Rodino, Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients,, J. Funct. Anal., 237 (2006), 634. doi: 10.1016/j.jfa.2005.12.017. Google Scholar [8] M. Cappiello, T. Gramchev and L. Rodino, Semilinear pseudo-differential equations and travelling waves,, Fields Institute Communications, 52 (2007), 213. Google Scholar [9] M. Cappiello, T. Gramchev and L. Rodino, Sub-exponential decay and uniform holomorphic extensions for semilinear pseudodifferential equations,, Comm. Partial Differential Equations, 35 (2010), 846. doi: 10.1080/03605300903509120. Google Scholar [10] M. Cappiello, T. Gramchev and L. Rodino, Entire extensions and exponential decay for semilinear elliptic equations,, J. Anal. Math., 111 (2010), 339. doi: 10.1007/s11854-010-0021-4. Google Scholar [11] M. Cappiello, T. Gramchev and L. Rodino, Decay estimates for solutions of nonlocal semilinear equations,, Nagoya Math. J., 218 (2015), 175. doi: 10.1215/00277630-2891745. Google Scholar [12] M. Cappiello and F. Nicola, Holomorphic extension of solutions of semilinear elliptic equations,, Nonlinear Anal., 74 (2011), 2663. doi: 10.1016/j.na.2010.12.021. Google Scholar [13] M. Cappiello and F. Nicola, Regularity and decay of solutions of nonlinear harmonic oscillators,, Adv. Math., 229 (2012), 1266. doi: 10.1016/j.aim.2011.10.018. Google Scholar [14] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions. Vols. I, II. Based,, in part, (1953). Google Scholar [15] G. Schiano, Perturbazioni non Lineari Per Alcuni Moltiplicatori di Fourier,, Master Thesis at University of Turin, (2012). Google Scholar [16] I. M. Gelfand and G. E. Shilov, Generalized Functions I,, Academic Press, (1964). Google Scholar [17] K. Gröchenig, Foundation of Time-frequency Analysis,, Birkhäuser, (2001). doi: 10.1007/978-1-4612-0003-1. Google Scholar [18] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. I and Vol. III,, Springer-Verlag, (1985). Google Scholar [19] F. Linares, D. Pilod and G. Ponce, Well-posedness for a higher-order Benjamin-Ono equation,, J. Differential Equations, 250 (2011), 450. doi: 10.1016/j.jde.2010.08.022. Google Scholar [20] M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation,, Nonlinear Anal., 51 (2002), 1073. doi: 10.1016/S0362-546X(01)00880-X. Google Scholar [21] L. Molinet, J. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations,, SIAM J. Math. Anal., 33 (2001), 982. doi: 10.1137/S0036141001385307. Google Scholar [22] H. Ono, Algebraic solitary waves in stratified fluids,, J. Phys. Soc. Japan, 39 (1975), 1082. doi: 10.1143/JPSJ.39.1082. Google Scholar [23] E. Stein, Harmonic Analysis,, Princeton University Press, (1993). Google Scholar [24] L. Schwartz, Théorie Des Distributions,, Hermann 1966, (1966). Google Scholar [25] T. Tao, Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbbR)$,, J. Hyperbolic Differ. Equ., 1 (2004), 27. doi: 10.1142/S0219891604000032. Google Scholar [26] G. N. Watson, A Treatise on the Theory of Bessel Functions,, Cambridge University Press, (1995). Google Scholar [27] M. Taylor, Pseudodifferential Operators and Nonlinear PDE,, Birkhäuser, (1991). doi: 10.1007/978-1-4612-0431-2. Google Scholar [28] M. Taylor, Partial Differential Equations, Vol. III,, Springer, (1996). doi: 10.1007/978-1-4684-9320-7. Google Scholar

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##### References:
 [1] C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation - a nonlinear Neumann problem in the plane,, Acta Math., 167 (1991), 107. doi: 10.1007/BF02392447. Google Scholar [2] T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, J. Fluid Mech., 29 (1967), 559. doi: 10.1017/S002211206700103X. Google Scholar [3] H. A. Biagioni and T. Gramchev, Fractional derivative estimates in Gevrey classes, global regularity and decay for solutions to semilinear equations in $\mathbbR^n$,, J. Differential Equations, 194 (2003), 140. doi: 10.1016/S0022-0396(03)00197-9. Google Scholar [4] J. Bona and Y. Li, Analyticity of solitary-wave solutions of model equations for long waves., SIAM J. Math. Anal., 27 (1996), 725. doi: 10.1137/0527039. Google Scholar [5] J. Bona and Y. Li, Decay and analyticity of solitary waves,, J. Math. Pures Appl., 76 (1997), 377. doi: 10.1016/S0021-7824(97)89957-6. Google Scholar [6] N. Burq and F. Planchon, On well-posedness for the Benjamin-Ono equation,, Math. Ann., 340 (2008), 497. doi: 10.1007/s00208-007-0150-y. Google Scholar [7] M. Cappiello, T. Gramchev and L. Rodino, Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients,, J. Funct. Anal., 237 (2006), 634. doi: 10.1016/j.jfa.2005.12.017. Google Scholar [8] M. Cappiello, T. Gramchev and L. Rodino, Semilinear pseudo-differential equations and travelling waves,, Fields Institute Communications, 52 (2007), 213. Google Scholar [9] M. Cappiello, T. Gramchev and L. Rodino, Sub-exponential decay and uniform holomorphic extensions for semilinear pseudodifferential equations,, Comm. Partial Differential Equations, 35 (2010), 846. doi: 10.1080/03605300903509120. Google Scholar [10] M. Cappiello, T. Gramchev and L. Rodino, Entire extensions and exponential decay for semilinear elliptic equations,, J. Anal. Math., 111 (2010), 339. doi: 10.1007/s11854-010-0021-4. Google Scholar [11] M. Cappiello, T. Gramchev and L. Rodino, Decay estimates for solutions of nonlocal semilinear equations,, Nagoya Math. J., 218 (2015), 175. doi: 10.1215/00277630-2891745. Google Scholar [12] M. Cappiello and F. Nicola, Holomorphic extension of solutions of semilinear elliptic equations,, Nonlinear Anal., 74 (2011), 2663. doi: 10.1016/j.na.2010.12.021. Google Scholar [13] M. Cappiello and F. Nicola, Regularity and decay of solutions of nonlinear harmonic oscillators,, Adv. Math., 229 (2012), 1266. doi: 10.1016/j.aim.2011.10.018. Google Scholar [14] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions. Vols. I, II. Based,, in part, (1953). Google Scholar [15] G. Schiano, Perturbazioni non Lineari Per Alcuni Moltiplicatori di Fourier,, Master Thesis at University of Turin, (2012). Google Scholar [16] I. M. Gelfand and G. E. Shilov, Generalized Functions I,, Academic Press, (1964). Google Scholar [17] K. Gröchenig, Foundation of Time-frequency Analysis,, Birkhäuser, (2001). doi: 10.1007/978-1-4612-0003-1. Google Scholar [18] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. I and Vol. III,, Springer-Verlag, (1985). Google Scholar [19] F. Linares, D. Pilod and G. Ponce, Well-posedness for a higher-order Benjamin-Ono equation,, J. Differential Equations, 250 (2011), 450. doi: 10.1016/j.jde.2010.08.022. Google Scholar [20] M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation,, Nonlinear Anal., 51 (2002), 1073. doi: 10.1016/S0362-546X(01)00880-X. Google Scholar [21] L. Molinet, J. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations,, SIAM J. Math. Anal., 33 (2001), 982. doi: 10.1137/S0036141001385307. Google Scholar [22] H. Ono, Algebraic solitary waves in stratified fluids,, J. Phys. Soc. Japan, 39 (1975), 1082. doi: 10.1143/JPSJ.39.1082. Google Scholar [23] E. Stein, Harmonic Analysis,, Princeton University Press, (1993). Google Scholar [24] L. Schwartz, Théorie Des Distributions,, Hermann 1966, (1966). Google Scholar [25] T. Tao, Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbbR)$,, J. Hyperbolic Differ. Equ., 1 (2004), 27. doi: 10.1142/S0219891604000032. Google Scholar [26] G. N. Watson, A Treatise on the Theory of Bessel Functions,, Cambridge University Press, (1995). Google Scholar [27] M. Taylor, Pseudodifferential Operators and Nonlinear PDE,, Birkhäuser, (1991). doi: 10.1007/978-1-4612-0431-2. Google Scholar [28] M. Taylor, Partial Differential Equations, Vol. III,, Springer, (1996). doi: 10.1007/978-1-4684-9320-7. Google Scholar
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