Some asymptotic limits for solutions of Burgers equation

§1 – Introduction. In this paper, we compute the limits (1) γ p = lim t → ∞ t 1 2 (1 − 1 p) u(·, t) p , 1 ≤ p ≤ ∞ for solutions u(·, t) of the equation (2) u t + a u x + b u u x = c u xx satisfying the Cauchy condition (3) u(x, 0) = u 0 (x), u 0 ∈ L 1 (R), that is, u(·, t) − u 0 1 → 0 as t → 0, t > 0. Here, u(·, t) p denotes the L p norm of u(·, t) as a function of x for fixed t, i.e., (4) u(·, t) p = +∞ −∞ | u(x, t) | p dx 1/p if 1 ≤ p < ∞, and (5) u(·, t) ∞ = sup x ∈ R | u(x, t) | for p = ∞. In equation (2) above, a, b, c are real constants, with c > 0. When b = 0 we have the familiar heat equation; our main concern is the case b = 0, the so-called Burgers equation [ 1 ], [ 3 ]. Using the Hopf-Cole transformation [ 4 ], [ 5 ], it is well known that the solution in this case is given by (6) u(x, t) = 1 √ 4 π c t 1 ϕ(x, t) +∞ −∞ e − (x − y − a t) 2 4 c t ϕ 0 (y) u 0 (y) dy, with (7) ϕ(x, t) = 1 √ 4 π c t +∞ −∞ e − (x − y − a t) 2 4 c t ϕ 0 (y) dy,