Non-Existence of Solutions for a Fourth-Order Structurally Damped Equation Under Nonlinear Memory Effect

Abstract

In this paper, we study a mixed wave-plate equation with rotational inertia, fractional damping and memory non-linearity. This research is a non-existence counterpart to a paper by D'Abbicco and Longen, in search of the critical exponent for the global in-time existence of small data solutions to the Cauchy problem for a mixed wave-plate equation, with a rotational inertia and a non-integer dissipation term and a memory-type non-linearity. We employ a modified test function argument to show that there are arbitrarily small initial data for which there are no solutions for the problem in the subcritical case. Additionally, we show non-existence of global solutions in the critical case when the memory exponent is greater than (n-2)/n.