Asymptotic Profiles of Solutions for Structural Damped Wave Equations

By Ryo Ikehata, Hiroshi Takeda. Journal of Dynamics and Differential Equations, 2016

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In this paper, we obtain several asymptotic profiles of solutions to the Cauchy problem for structurally damped wave equations $$\partial _{t}^{2} u - \varDelta u + \nu (-\varDelta )^{\sigma } \partial _{t} u=0$$∂t2u-Δu+ν(-Δ)σ∂tu=0, where $$\nu >0$$ν>0 and $$0< \sigma \le 1$$0<σ≤1. Our result is the approximation formula of the solution by a constant multiple of a special function as $$t \rightarrow \infty $$t→∞, which states that the asymptotic profiles of the solutions are classified into 5 patterns depending on the values $$\nu $$ν and $$\sigma $$σ. Here we emphasize that our main interest of the paper is in the case $$\sigma \in (\frac{1}{2},1)$$σ∈(12,1).

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

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