    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:
  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  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  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  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  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  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  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  M. Cappiello, T. Gramchev and L. Rodino, Semilinear pseudo-differential equations and travelling waves, Fields Institute Communications, 52 (2007), 213-238. Google Scholar  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  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  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  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  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  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  G. Schiano, Perturbazioni non Lineari Per Alcuni Moltiplicatori di Fourier, Master Thesis at University of Turin, 2012. Google Scholar  I. M. Gelfand and G. E. Shilov, Generalized Functions I, Academic Press, New York and London, 1964. Google Scholar  K. Gröchenig, Foundation of Time-frequency Analysis, Birkhäuser, 2001. doi: 10.1007/978-1-4612-0003-1.  Google Scholar  L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. I and Vol. III, Springer-Verlag, 1985. Google Scholar  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  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  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  H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082.  Google Scholar  E. Stein, Harmonic Analysis, Princeton University Press, 1993. Google Scholar  L. Schwartz, Théorie Des Distributions, Hermann 1966, Paris. Google Scholar  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  G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995. Google Scholar  M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, 1991. doi: 10.1007/978-1-4612-0431-2.  Google Scholar  M. Taylor, Partial Differential Equations, Vol. III, Springer, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

show all references

##### References:
  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  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  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  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  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  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  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  M. Cappiello, T. Gramchev and L. Rodino, Semilinear pseudo-differential equations and travelling waves, Fields Institute Communications, 52 (2007), 213-238. Google Scholar  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  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  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  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  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  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  G. Schiano, Perturbazioni non Lineari Per Alcuni Moltiplicatori di Fourier, Master Thesis at University of Turin, 2012. Google Scholar  I. M. Gelfand and G. E. Shilov, Generalized Functions I, Academic Press, New York and London, 1964. Google Scholar  K. Gröchenig, Foundation of Time-frequency Analysis, Birkhäuser, 2001. doi: 10.1007/978-1-4612-0003-1.  Google Scholar  L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. I and Vol. III, Springer-Verlag, 1985. Google Scholar  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  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  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  H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082.  Google Scholar  E. Stein, Harmonic Analysis, Princeton University Press, 1993. Google Scholar  L. Schwartz, Théorie Des Distributions, Hermann 1966, Paris. Google Scholar  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  G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995. Google Scholar  M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, 1991. doi: 10.1007/978-1-4612-0431-2.  Google Scholar  M. Taylor, Partial Differential Equations, Vol. III, Springer, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar
  Amin Esfahani, Steve Levandosky. Solitary waves of the rotation-generalized Benjamin-Ono equation. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 663-700. doi: 10.3934/dcds.2013.33.663  Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215  Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941  Jerry Bona, H. Kalisch. Singularity formation in the generalized Benjamin-Ono equation. Discrete & Continuous Dynamical Systems, 2004, 11 (1) : 27-45. doi: 10.3934/dcds.2004.11.27  Kenta Ohi, Tatsuo Iguchi. A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1205-1240. doi: 10.3934/dcds.2009.23.1205  Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051  Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237  Lufang Mi, Kangkang Zhang. Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 689-707. doi: 10.3934/dcds.2014.34.689  G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327  Robert Schippa. On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5189-5215. doi: 10.3934/dcds.2020225  Eddye Bustamante, José Jiménez Urrea, Jorge Mejía. The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1177-1203. doi: 10.3934/cpaa.2019057  Luc Molinet, Francis Ribaud. Well-posedness in $H^1$ for generalized Benjamin-Ono equations on the circle. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1295-1311. doi: 10.3934/dcds.2009.23.1295  Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583  Alan Compelli, Rossen Ivanov. Benjamin-Ono model of an internal wave under a flat surface. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4519-4532. doi: 10.3934/dcds.2019185  José R. Quintero, Alex M. Montes. Exact controllability and stabilization for a general internal wave system of Benjamin-Ono type. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021021  Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15  Yaping Wu, Xiuxia Xing, Qixiao Ye. Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 47-66. doi: 10.3934/dcds.2006.16.47  Yiren Chen, Zhengrong Liu. The bifurcations of solitary and kink waves described by the Gardner equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1629-1645. doi: 10.3934/dcdss.2016067  H. Kalisch. Stability of solitary waves for a nonlinearly dispersive equation. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 709-717. doi: 10.3934/dcds.2004.10.709  Boling Guo, Zhaohui Huo. The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in $L^2$. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 121-136. doi: 10.3934/dcds.2006.16.121

2020 Impact Factor: 1.392