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J. Evol. Equ. Journal of Evolution © 2019 Springer Nature Switzerland AG Equations https://doi.org/10.1007/s00028-019-00554-0 A note on the Moore–Gibson–Thompson equation with memory of type II Filippo Dell’Oro, Irena Lasiecka and Vittorino Pata Abstract. We consider the Moore–Gibson–Thompson equation with memory of type II ∂ u(t ) + α∂ u(t ) + β A∂ u(t ) + γ Au(t ) − g(t − s)A∂ u(s)ds = 0 ttt tt t t where A is a strictly positive selfadjoint linear operator (bounded or unbounded) and α, β, γ > 0 satisfy the relation γ ≤ αβ. First, we prove well-posedness of finite energy solutions, without requiring any restriction on the total mass of g. This extends previous results in the literature, where such a restriction was imposed. Second, we address an open question within the context of longtime behavior of solutions. We show that an “overdamping” in the memory term can destabilize the originally stable dynamics. In fact, it is always possible to find memory kernels g, complying with the usual mass restriction < β, such that the equation admits solutions with energy growing exponentially fast, even in the regime γ< αβ where the corresponding model without memory is exponentially
Journal of Evolution Equations – Springer Journals
Published: Dec 4, 2019
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