Finding the exact decay rate of all solutions to some second order evolution equations with dissipation

TL;DR

This paper precisely characterizes the decay rates of solutions to certain second order evolution equations with damping, revealing the coexistence of slow and fast decay behaviors based on spectral properties.

Contribution

It establishes the exact decay rates for all solutions, distinguishing between slow and fast solutions in a general abstract setting with applications to hyperbolic equations.

Findings

Slow solutions decay as negative powers of time.

Fast solutions decay exponentially.

Decay rates depend on spectral properties of the operator A.

Abstract

We consider an abstract second order evolution equation with damping. The "elastic" term is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We investigate the asymptotic behavior of solutions. We prove the coexistence of slow solutions and fast solutions. Slow solutions live close to the kernel of A, and decay as negative powers of t as solutions of the first order equation obtained by neglecting the operator A and the second order time-derivatives in the original equation. Fast solutions live close to the range of A and decay exponentially as solutions of the linear homogeneous equation obtained by neglecting the nonlinear terms in the original equation. The abstract results apply to semilinear dissipative hyperbolic equations.