Exponential Energy Decay for Damped Klein–Gordon Equation with Nonlinearities of Arbitrary Growth

Abstract

We derive a uniform exponential decay of the total energy for the nonlinear Klein-Gordon equation with a damping around spatial infinity in R N or in the exterior of a star-shaped obstacle. Such a result was first proved by Zuazua [40,41] for defocusing nonlinearity with moderate growth, and later extended to the energy subcritical case by Dehman-Lebeau-Zuazua [7], using linear approximation and unique continuation arguments. We propose a different approach based solely on Morawetz-type a priori estimates, which applies to defocusing nonlinearity of arbitrary growth, including the energy critical case, the supercritical case and exponential nonlinearities in any dimensions. One advantage of our proof, even in the case of moderate growth, is that the decay rate is independent of the nonlinearity. We can also treat the focusing case for those solutions with energy less than the one of the ground state, once we get control of the nonlinear part in Morawetz-type estimates. In particular this can be achieved when we have the scattering for the undamped equation.

Boundary stabilization of the wave and Schrödinger equations in exterior domains

“…In [23] Zuazua considered the nonlinear Klein-gordon equations with dissipative term and he proved the exponential decay of energy through the weighted energy method. This result has been generalized by Aloui et al [5] for more general nonlinearities. We refer the reader to the works of Dehman et al [9] and Laurent et al [14] for related results.…”

Energy decay for linear dissipative wave equations in exterior domains

“…where (u 0 , u 1 ) ∈ (H 1 L m ) × (L 2 L m ) and m ∈ [1,2]. It is worth noticing that the decay rate given in (1.9) is the same one for the heat equation.…”

On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

“…We also mention the recent result of Aloui, Ibrahim and Nakanishi [1] for R d . Their method of proof is very different and uses Morawetz-type estimates.…”

“…To illustrate the problem, let us take the examples of a sequence with two linear profiles p (1) n and p (2) n , and denote q (1) n , q (2) n the associated nonlinear profiles, solutions of the nonlinear equation with same initial data. We want to prove that the solution of u n + u n = u 5 n with initial data p (1) n + p (2) n at time 0 can be approximately written u n ≈ q (1) n + q (2) n in energy.…”

“…We want to prove that the solution of u n + u n = u 5 n with initial data p (1) n + p (2) n at time 0 can be approximately written u n ≈ q (1) n + q (2) n in energy. Because both of them have the same initial data, a boot strap argument allows to conclude if we prove that q…”