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We consider an initial boundary value problem of the multi-dimensionalBoussinesq equations in the absence of thermal diffusion with velocity dampingor velocity diffusion under the stress free boundary condition in horizontallyperiodic strip domain. We prove the global-in-time existence of classicalsolutions in high order Sobolev spaces satisfying high order compatibilityconditions around the linearly stratified equilibrium, the convergence of thetemperature to the asymptotic profile, and sharp decay rates of the velocityfield and temperature fluctuation in all intermediate norms based on spectralanalysis combined with energy estimates. To the best of our knowledge, ourresults provide first sharp decay rates for the temperature fluctuation and thevertical velocity to the linearly stratified Boussinesq equations in allintermediate norms.

Asymptotic stability and sharp decay rates to the linearly stratified Boussinesq equations in horizontally periodic strip domain