Re
cently, a time-de
cay framework
class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300882&_mathId=si1.gif&_user=111111111&_pii=S0022039616300882&_rdoc=1&_issn=00220396&md5=4468a8f0f57ba30c9b3cfa257dc82dac">
class="imgLazyJSB inlineImage" height="23" width="178" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616300882-si1.gif">cript>
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[49] for linearized dissipative hyperboli
c systems, whi
ch allows to pay less attention to the traditional spe
ctral analysis. However, owing to interpolation te
chniques, those de
cay results for nonlinear hyperboli
c systems hold true only in higher dimensions
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300882&_mathId=si2.gif&_user=111111111&_pii=S0022039616300882&_rdoc=1&_issn=00220396&md5=a834e445bf935139700af8d13d69aa9a" title="Click to view the MathML source">(n≥3)class="mathContainer hidden">class="mathCode">, and the analysis in low dimensions (say,
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300882&_mathId=si3.gif&_user=111111111&_pii=S0022039616300882&_rdoc=1&_issn=00220396&md5=21f68670c4a868e471adce645b78ba45" title="Click to view the MathML source">n=1,2class="mathContainer hidden">class="mathCode">) was left open. We try to give a satisfa
ctory answer in the
current work. First of all, we develop new time-de
cay properties on the frequen
cy-lo
calization
Duhamel prin
ciple, and then it is shown that the
classi
cal solution and its derivatives of fra
ctional order de
cay at the optimal algebrai
c rate in dimensions
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300882&_mathId=si3.gif&_user=111111111&_pii=S0022039616300882&_rdoc=1&_issn=00220396&md5=21f68670c4a868e471adce645b78ba45" title="Click to view the MathML source">n=1,2class="mathContainer hidden">class="mathCode">, by using a new te
chnique whi
ch is the so-
called “pie
cewise Duhamel prin
ciple” in lo
calized time-weighted energy approa
ches
compared to
[49]. Finally, as dire
ct appli
cations, expli
cit de
cay statements are worked out for some relevant examples subje
cted to the same dissipative stru
cture, for instan
ce, damped
compressible Euler equations, the thermoelasti
city with se
cond sound, and Timoshenko systems with equal wave speeds.