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Thresholds for low regularity solutions to wave equations with structural damping. (English) Zbl 1458.35054

Summary: We study the asymptotic behavior of solutions to wave equations with the structural damping term \[ u_{t t} - \Delta u + \Delta^2 u_t = 0,\quad u(0, x) = u_0(x), u_t(0, x) = u_1(x), \] in the whole space. New thresholds are reported in this paper that indicate which of the diffusion wave property and the non-diffusive structure dominates in low regularity cases. We develop to that end the previous authors’ research [J. Hyperbolic Differ. Equ. 17, No. 3, 569–589 (2020; Zbl 1455.35017)] where they have proposed a threshold that expresses whether the parabolic-like property or the wave-like property strongly appears in the solution to some regularity-loss type dissipative wave equation.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L30 Initial value problems for higher-order hyperbolic equations

Citations:

Zbl 1455.35017
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References:

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