Well-posedness for the Cauchy problem of spatially weighted dissipative equation

TL;DR

This paper studies the well-posedness of the Cauchy problem for a spatially weighted dissipative equation, introducing a generalized Hankel Transform and applying a specialized Young's Inequality to establish space-time estimates.

Contribution

It introduces a generalized Hankel Transform and a special Young's Inequality to analyze the well-posedness of weighted dissipative equations.

Findings

Established space-time estimates for the equation

Proved well-posedness in weighted Lebesgue spaces

Developed analytical solutions using the generalized Hankel Transform

Abstract

This paper mainly investigates the Cauchy problem of the spatially weighted dissipative equation with initial data in the weighted Lebesgue space. A generalized Hankel Transform is introduced to derive the analytical solution and a special Young's Inequality has been applied to prove the space-time estimates for this type of equation.