The σ‐evolution equations with friction and unbounded coefficients

Abstract

We present in this article the study on the decay estimates for solutions of the Cauchy problem for a class of linear ‐evolution equations with unbounded coefficients with in space dimension , where the coefficient is continuous and unbounded satisfying the asymptotic behavior as , with the parameter , and is a positive and bounded continuous function. By investigating the diffusion phenomenon, we obtain new results that generalize our previous study for the damped wave models with constant and nearly constant coefficients. Using the decay estimates for the solution of the linear problem, we establish a result on the global (in time) existence of the small data solution for the corresponding semilinear Cauchy problem with the nonlinearity of power type.