On the MGT equation with memory of type II
Abstract
We consider the MooreGibsonThompson equation with memory of type II $$ \partial_{ttt} u(t) + \alpha \partial_{tt} u(t) + \beta A \partial_t u(t) + \gamma Au(t)\int_0^t g(ts) A \partial_t u(s){\rm d} s=0 $$ where $A$ is a strictly positive selfadjoint linear operator (bounded or unbounded) and $\alpha,\beta,\gamma>0$ satisfy the relation $\gamma\leq\alpha\beta$. First, we prove a wellposedness result without requiring any restriction on the total mass $\varrho$ of $g$. Then we show that it is always possible to find memory kernels $g$, complying with the usual mass restriction $\varrho<\beta$, such that the equation admits solutions with energy growing exponentially fast. In particular, this provides the answer to a question raised in "F. Dell'Oro, I. Lasiecka, V. Pata, J. Differential Equations 261 (2016), 41884222".
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.08203
 Bibcode:
 2019arXiv190408203D
 Keywords:

 Mathematics  Analysis of PDEs