The dominant asymptotic term of solution to the Hamer equation

Abstract

It is shown that the dominant asymptotic term of solution to the problem

u_t+uu_x=K*u−u, u|_t=0=u₀,

where K(x)=(1/2)e^−|x|, is a function u₀(x,t)=2ζ(x,t)(1−∫_−∞^xζ(s,t) ds)⁻¹

with ζ(x,t)=M₀(h)e^x2/4t/√4πt, M₀(h)=∫_−∞^∞h(s) ds and h(x)=(1/2)u₀(x)exp {−(1/2)∫_−∞^xu₀(s) ds}, and the function u₀(x,t) coincides with the principal asymptotic term of solution to the Burgers equation with initial function u₀.