Longtime behavior of the semilinear wave equation with gentle dissipation

Abstract

The paper investigates the well-posedness and longtime dynamics of the semilinear wave equation with gentle dissipation utt - ▵u + γ(-▵)αut + f(u) = g(x), with α ∈ (0, 1/2). The main results are concerned with the relationships among the growth exponent p of nonlinearity f(u) and the well-posedness and longtime behavior of solutions of the equation. We show that (i) the well-posedness and longtime dynamics of the equation are of characters of parabolic equations as 1 ≤ p