Diffusion phenomenon of solutions for the damped wave equation

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The solution to the Cauchy problem for the damped wave equation behaves like that for the corresponding diffusive equation as time tends to infinity, which is called the diffusion phenomenon of solutions to the damped wave equation. When the damping term with coefficients depending on time or space variables is effective, the Cauchy problems for the semilinear damped wave equations are investigated, and the critical exponents on the semilinear term have been explored. In the corresponding supercritical and subcritical cases, the existence of global-in-time solution, its decay rate, asymptotic profile, blow-up of solution within a finite time etc. are considered.