Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

Abstract

<p style='text-indent:20px;'>Boundary feedback stabilization of a <i>critical</i> third–order (in time) semilinear Jordan–Moore–Gibson–Thompson (JMGT) is considered. The word <i>critical</i> here refers to the usual case where media–damping effects are non–existent or non–measurable and therefore cannot be relied upon for stabilization purposes. Motivated by modeling aspects in high-intensity focused ultrasound (HIFU) technology, the boundary feedback under consideration is supported only on a portion of the boundary. At the same time, the remaining part is undissipated and subject to Neumann/Robin boundary conditions. As such, unlike Dirichlet, it fails to satisfy the Lopatinski condition, a fact which compromises tangential regularity on the boundary [<xref ref-type="bibr" rid="b37">37</xref>]. In such a configuration, the analysis of uniform stabilization from the boundary becomes subtle and requires careful geometric considerations and microlocal analysis estimates. The nonlinear effects in the model demand construction of suitably small solutions which are invariant under the dynamics. The assumed smallness of the initial data is required only at the lowest energy level topology, which is sufficient to construct sufficiently smooth solutions to the nonlinear model.</p>