# 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, 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), 107-126. 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), 559-592. 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), 140-165. 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), 725-737. doi: 10.1137/0527039.  Google Scholar [5] J. Bona and Y. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430. 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), 497-542. 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), 634-654. 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), 213-238.  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), 846-877. 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), 339-367. 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), 175-198. 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), 2663-2681. 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), 1266-1299. 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, on notes left by Harry Bateman, McGraw-Hill Book Company, Inc., New York-Toronto-London, 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, New York and London, 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), 450-475. 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), 1073-1085. 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), 982-988. doi: 10.1137/S0036141001385307.  Google Scholar [22] H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. 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, Paris.  Google Scholar [25] T. Tao, Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbbR)$, J. Hyperbolic Differ. Equ., 1 (2004), 27-49. doi: 10.1142/S0219891604000032.  Google Scholar [26] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 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), 107-126. 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), 559-592. 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), 140-165. 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), 725-737. doi: 10.1137/0527039.  Google Scholar [5] J. Bona and Y. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430. 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), 497-542. 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), 634-654. 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), 213-238.  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), 846-877. 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), 339-367. 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), 175-198. 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), 2663-2681. 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), 1266-1299. 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, on notes left by Harry Bateman, McGraw-Hill Book Company, Inc., New York-Toronto-London, 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, New York and London, 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), 450-475. 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), 1073-1085. 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), 982-988. doi: 10.1137/S0036141001385307.  Google Scholar [22] H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. 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, Paris.  Google Scholar [25] T. Tao, Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbbR)$, J. Hyperbolic Differ. Equ., 1 (2004), 27-49. doi: 10.1142/S0219891604000032.  Google Scholar [26] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 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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