Uniqueness and asymptotic-behavior for some scalar convection-diffusion equations

Abstract

We prove the uniqueness of the fundamental entropy solutions u(x, y, t) of the equation: (R) u(t) - DELTA(x) u + partial derivative(y) (Absolute value of u q-1 u) = 0, R(n-1) x R x R+ when 1 < q < 1+(2/(n - 1)) if n > 2 and 1 < q less-than-or-equal-to 2 if n = 1, 2. As a consequence, we prove that the large time behaviour of solutions to the equation (CD) u(t) - DELTA(x) u - partial derivative(yy)2 u + partial derivative(y) (Absolute value of u q-1 u) = 0, R(n-1) x R x R+ with initial data u0 is-an-element-of L1 (R(n)) is given by the fundamental solutions of (R) with mass integral u0 when 1 < q < 1 + (1 /n). This completes a result by Escobedo, Vazquez and Zuazua for positive solutions.