In this short note, we want to describe the logarithmic convexity argument for third order in time partial differential equations. As a consequence, we first prove a uniqueness result whenever certain conditions on the parameters are satisfied. Later, we show the instability of the solutions if the initial energy is less or equal than zero.
HadamardJ.Sur les problèmes aux dérivées partielles et leur signification physique. Princet Univ Bull1902; XIII(4): 49–52.
2.
KnopsRJ. Instability and the ill-posed Cauchy problem in elasticity. In: HopkingsHSewellM (eds.) Mechanics of solids: the Rodney Hill 60th anniversary. Berlin: Elsevier, 1982, pp. 357–382.
3.
KnopsRJPayneLE.Growth estimates for solutions of evolutionary equations in Hilbert spaces with applications to elastodynamics. Arch Rat Mech Anal1971; 41: 363–398.
4.
QuintanillaRStraughanB.Growth and uniqueness in thermoelasticity. Proc Roy Soc A Math Phys Eng Sci2000; 456: 1419–1430.
5.
WilkesNS.Continuous dependence and instability in linear thermoelasticity. SIAM J Math Anal1980; 11(2): 292–299.
6.
PellicerMQuintanillaR.On uniqueness and instability for some thermomechanical problems involving the Moore–Gibson–Thompson equation. Z Angew Math Phys2020; 71: 84.
7.
FernándezJRQuintanillaR. Uniqueness for a high order ill posed problem. Proc Roy Soc Edim Sect A Math. Epub ahead of print 15 July 2022, https://doi.org/10.1017/prm.2022.46
8.
MagañaRQuintanillaR.On the existence and uniqueness in phase-lag thermoelasticity. Meccanica2018; 53: 125–134.
9.
LebedevLPGladwellGML. On spatial effects of modelling in linear viscoelasticity. J Elasticity1997; 47: 241–250.
