Asymptotic stability of diffusion wave for a semilinear wave equation with damping

This paper studies the asymptotic behavior of the solution to the Cauchy problem of a semilinear wave equation with damping v(tt)+ v(t)+ f(Dv) = Delta v, x is an element of R-n, under some smallness conditions. By applying elementary energy method, we prove the solution of the above equation tends to the planar diffusion wave (v) over bar (x(1)/root 1+t) timeasymptotically, where (v) over bar (x(1)/root 1+t) is a self-similar solution of the one dimensional equation (v) over bar + C-0(v) over bar (2)(x1) = (v) over bar (x1x1), (v) over bar(+/-infinity, t) = v(+/-), v(+) not equal v(-), with C-0 = 1/2 partial derivative(2)f(xi)/partial derivative xi(2)(1)vertical bar xi=0. In addition, this paper gives the L-infinity time decay rate, namely, parallel to v - (v) over bar parallel to(L infinity) = O(1)epsilon(2)(1 + t)gamma/4, where gamma = min{3, n}. (C) 2021 Published by Elsevier Inc.