Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption

By Kenji Nishihara, Huijiang Zhao. 2006

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Abstract We consider the Cauchy problem for the damped wave equation with absorption u t t − Δ u + u t + | u | p − 1 u = 0 , ( t , x ) ∈ R + × R N . The behavior of u as t → ∞ is expected to be the same as that for the corresponding heat equation ϕ t − Δ ϕ + | ϕ | p − 1 ϕ = 0 , ( t , x ) ∈ R + × R N , which has the similarity solution w a ( t , x ) with the form t − 1 / ( p − 1 ) f ( x / t ) depending on a = lim | x | → ∞ | x | 2 / ( p − 1 ) f ( x ) ⩾ 0 provided that p is less than the Fujita exponent p c ( N ) : = 1 + 2 / N . In this paper, as a first step, if 1 p p c ( N ) and the data ( u 0 , u 1 ) ( x ) decays exponentially as | x | → ∞ without smallness condition, the solution is shown to decay with rates as t → ∞ , (∗) ( ‖ u ( t ) ‖ L 2 , ‖ u ( t ) ‖ L p + 1 , ‖ ∇ u ( t ) ‖ L 2 ) = O ( t − 1 p − 1 + N 4 , t − 1 p − 1 + N 2 ( p + 1 ) , t − 1 p − 1 − 1 2 + N 4 ) , those of which seem to be reasonable, because the similarity solution w a ( t , x ) has the same decay rates as (∗) . For the proof, the weighted L 2 -energy method will be employed with suitable weight, similar to that in Todorova and Yordanov [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464–489].

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/39d0014f68c26055081fac83dde2a7848d9ee44e

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Citations in the corpus

Damped wave equation with a critical nonlinearity (2005, background): “Recently in [15], sharp L-time decay estimates of solutions to the nonlinear damped wave equation (1.”
Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption (2006, methodology): “The first one is to obtain the decay properties in the subcritical case for the data in reasonable weighted space, which extends the initial data class in Nishihara and Zhao [29].”