2021 Volume 73 Issue 4 Pages 1091-1128
This paper is concerned with supersolutions to parabolic equations with space-dependent diffusion coefficients. Given the behavior of the diffusion coefficient with polynomial order at spatial infinity, a family of supersolutions with slowly decaying property at spatial infinity is provided. As a first application, weighted 𝐿2 type decay estimates for the initial-boundary value problem of the parabolic equation are proved. The second application is the study of the exterior problem of wave equations with space-dependent damping terms. By using supersolution provided above, energy estimates with polynomial weight and diffusion phenomena are shown. There is a slight improvement compared to the previous work about the assumption of the initial data.
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