Remarks on the asymptotic behavior of solutions to damped evolution equations in Hilbert space

Bloom, Frederick

doi:10.1090/S0002-9939-1979-0529206-7

Remarks on the asymptotic behavior of solutions to damped evolution equations in Hilbert space

Author: Frederick Bloom
Journal: Proc. Amer. Math. Soc. 75 (1979), 25-31
MSC: Primary 34G10; Secondary 35B40
DOI: https://doi.org/10.1090/S0002-9939-1979-0529206-7
MathSciNet review: 529206
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Abstract: Lower bounds are derived for the norms of solutions to a class of intitial-value problems associated with the damped evolution equation ${u_{tt}} + A{u_t} + Bu = 0$ in Hilbert space. Under appropriate assumptions on the linear operator B it is shown that even in the special strongly damped case where $A = \Gamma I,\Gamma > 0$ , solutions are bounded away from zero as $t \to + \infty$ , even when $\Gamma \to + \infty$ .

[1] C. M. Dafermos, Wave equations with weak damping, SIAM J. Appl. Math. 18 (1970), 759-767. MR 0262711 (41:7316)
[2] D. L. Russell, Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods, J. Differential Equations 19 (1975), 344-370. MR 0425291 (54:13248)
[3] H. A. Levine, Uniqueness and growth of weak solutions to certain linear differential equations in Hilbert space, J. Differential Equations 17 (1975), 73-81. MR 0358014 (50:10479)
[4] R. J. Knops and L. E. Payne, Growth estimates for solutions of evolutionary equations in Hilbert space with applications in elastodynamics, Arch. Rational Mech. Anal. 41 (1971), 363-398. MR 0330731 (48:9068)
[5] H. A. Levine, Logarithmic convexity, first order differential inequalities and some applications, Trans. Amer. Math. Soc. 152 (1970), 299-320. MR 0274988 (43:746)
[6] -, Logarithmic convexity and the Cauchy problem for some abstract second order differential inequalities, J. Differential Equations 8 (1970), 34-55. MR 0259303 (41:3945)
[7] H. A. Levine and L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations 16 (1974), 319-334. MR 0470481 (57:10235)
[8] -, Some nonexistence theorems for initial-boundary value problems with nonlinear boundary constraints, Proc. Amer. Math. Soc. 46 (1974), 277-284. MR 0364841 (51:1095)
[9] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $\rho u = - Au + \mathcal{F}(u)$ , Trans. Amer. Math. Soc. 192 (1974), 1-21. MR 0344697 (49:9436)
[10] -, Some nonexistence and instability theorems for formally parabolic equations of the form $\rho {u_t} = Au + \mathcal{F}(u)$ , Arch. Rational Mech. Anal. 51 (1973), 371-386. MR 0348216 (50:714)