DIFFUSION PHENOMENA OF SOLUTIONS TO THE CAUCHY PROBLEM FOR THE DAMPED WAVE EQUATIONS

摘要

In this essay we consider the Cauchy problem for the damped wave equation The equation is also called the telegraph equation or wave equation with dissipation. When f(u) ≡ 0, it is, of course, the linear damped wave equation, and when the semilinear prblem is considered, the semilinear term f(u) is assumed to be with the exponent p > 1. Many mathematicians have recognized that "The solution of the damped wave equation has the diffusion phenomena This means that the solution u(t, x) of (DW) behaves like the solution ?(t, x) as t ∞ to the Cauchy problem for the corresponding heat equation or diffusion equation In recent decades the author has investigated this problem and the related ones, which are discussed in this essay.