Electron. J. Differential Equations,
Vol. 2019 (2019), No. 91, pp. 116.
Decay estimates for nonlinear wave equations with variable coefficients
Michael Roberts
Abstract:
We study the longtime behavior of solutions to a particular class of nonlinear
wave equations that appear in models for waves traveling in a nonhomogeneous
gas with variable damping. Specifically, decay estimates for the energy of
such solutions are established.
We find three different regimes of energy decay depending on the
exponent of the absorption term
and show the existence of two
critical exponents
and
.
For
,
the decay of solutions of the nonlinear equation
coincides with that of the corresponding linear problem.
For
,
the solution decays much faster. The other
critical exponent
further divides this region
into two subregions with different decay rates. Deriving the sharp decay of
solutions even for the linear problem with potential
is a delicate
task and requires serious strengthening of the multiplier method. Here we use
a modification of an approach of Todorova and Yordanov to derive the exact
decay of the nonlinear equation.
Submitted March 19, 2019. Published July 23, 2019.
Math Subject Classifications: 35B33, 35B40, 35L70.
Key Words: Energy estimates; dissipative nonlinear wave; subsolution;
approximate solution; nonlinear exponent.
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Michael Roberts
Department of Mathematics
North Dakota State University
Minard 408. 1210 Albrecht Boulevard
Fargo, ND 581086050, USA
email: michael.roberts.1@ndus.edu

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