The long-time behaviour of solutions is investigated for the porous medium equation with convection: \[ u_t- (u^m)_{xx}+ (u^q)_x= 0, \qquad \text{in } \mathbb{R}\times (0,+\infty), \] and initial data \(u_0\in L^1(R)\) when \(M=| u_0| _{L^1} > 0\). The case \(m>1\) and \(q>1\) is only considered. This type of equation arises from modelling the transport of a solute through a porous medium under the assumption that the solute undergoes equilibrium absorption with the porous matrix.
The well-posedness of the Cauchy problem to this equation has been established by Ph. Bénilan and H. Touré [in C. R. Ann. Inst. H. Poincaré, Anal. Non Linéaire 12, 727-761 (1995; Zbl 0839.35068)]. In the article under review, the authors work with mild solutions in the sense of the nonlinear semigroup theory. Their analysis shows that there is a critical value \(q_1=m+1\) such that: (i) if \(q>q_1\), the solution \(u\) behaves as \(t\to +\infty\) as the Barenblatt-Pattle similarity solution \(E_M\) to the porous medium equation; (ii) if \(q=q_1\), the solution \(u\) behaves as \(t\to +\infty\) as the source-type solution \(\Theta_M\) to the porous medium equation with convection; (iii) if \(q\in (1,q_1)\), the long-time dynamics is dominated by the convective part of the equation; the solution \(u\) behaves as \(t\to +\infty\) as the unique non-negative entropy solution \(F_M\) to \(F_{Mt}+ (F_M^q)_x= 0\).