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Decay estimates for solutions of nonlocal semilinear equations

Part of: Partial differential equations Equations of mathematical physics and other areas of application Elliptic equations and systems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  11 January 2016

Marco Cappiello ,
Todor Gramchev  and
Luigi Rodino
Marco Cappiello
Affiliation:
Dipartimento di Matematica, Università di Torino, 10123 Torino, Italy, marco.cappiello@unito.it
Todor Gramchev
Affiliation:
Dipartimento di Matematica e Informatica, Università di Cagliari, 09124 Cagliari, Italy, todor@unica.it
Luigi Rodino
Affiliation:
Dipartimento di Matematica, Università di Torino, 10123 Torino, Italy, luigi.rodino@unito.it
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Abstract

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We investigate the decay for |x|→∞ of weak Sobolev-type solutions of semilinear nonlocal equations Pu = F(u). We consider the case when P = p(D) is an elliptic Fourier multiplier with polyhomogeneous symbol p(ξ), and we derive algebraic decay estimates in terms of weighted Sobolev norms. Our basic example is the celebrated Benjamin–Ono equation

for internal solitary waves of deep stratified fluids. Their profile presents algebraic decay, in strong contrast with the exponential decay for KdV shallow water waves.

MSC classification

Secondary: 35J61: Semilinear elliptic equations 35B40: Asymptotic behavior of solutions 35S05: Pseudodifferential operators 35Q51: Soliton-like equations
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 218 , June 2015 , pp. 175 - 198
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

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