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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

M. Cappiello, T. Gramchev and L. Rodino, Semilinear pseudo-differential equations and travelling waves, Fields Institute Communications, 52 (2007), 213-238.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

G. Schiano, Perturbazioni non Lineari Per Alcuni Moltiplicatori di Fourier, Master Thesis at University of Turin, 2012.

[16]

I. M. Gelfand and G. E. Shilov, Generalized Functions I, Academic Press, New York and London, 1964.

[17]

K. Gröchenig, Foundation of Time-frequency Analysis, Birkhäuser, 2001. doi: 10.1007/978-1-4612-0003-1.

[18]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. I and Vol. III, Springer-Verlag, 1985.

[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.

[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.

[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.

[22]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082.

[23]

E. Stein, Harmonic Analysis, Princeton University Press, 1993.

[24]

L. Schwartz, Théorie Des Distributions, Hermann 1966, Paris.

[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.

[26]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995.

[27]

M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, 1991. doi: 10.1007/978-1-4612-0431-2.

[28]

M. Taylor, Partial Differential Equations, Vol. III, Springer, 1996. doi: 10.1007/978-1-4684-9320-7.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

M. Cappiello, T. Gramchev and L. Rodino, Semilinear pseudo-differential equations and travelling waves, Fields Institute Communications, 52 (2007), 213-238.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

G. Schiano, Perturbazioni non Lineari Per Alcuni Moltiplicatori di Fourier, Master Thesis at University of Turin, 2012.

[16]

I. M. Gelfand and G. E. Shilov, Generalized Functions I, Academic Press, New York and London, 1964.

[17]

K. Gröchenig, Foundation of Time-frequency Analysis, Birkhäuser, 2001. doi: 10.1007/978-1-4612-0003-1.

[18]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. I and Vol. III, Springer-Verlag, 1985.

[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.

[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.

[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.

[22]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082.

[23]

E. Stein, Harmonic Analysis, Princeton University Press, 1993.

[24]

L. Schwartz, Théorie Des Distributions, Hermann 1966, Paris.

[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.

[26]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995.

[27]

M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, 1991. doi: 10.1007/978-1-4612-0431-2.

[28]

M. Taylor, Partial Differential Equations, Vol. III, Springer, 1996. doi: 10.1007/978-1-4684-9320-7.

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