Asymptotic profiles in contaminant transport in porous media with variable diffusion (Q2747833)

In this paper the author studies the large time behavior of a reactive solute which undergoes equilibrium adsorbtion in a porous medium with variable diffusion; the main results are concerned with 1-D case. The transport model is the convection-diffusion problem NEWLINE\[NEWLINEu_t-\bigl(a(x)u_x \bigr)_x+ q |u|^{q-1} u_x=0\quad \text{in }(0,\infty) \times\mathbb{R},NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0,.)= u_0, NEWLINE\]NEWLINE where \(q>1\). Under some assumptions about the asymptotic behavior of the coefficient \(a(x)\), a precise asymptotic behavior of a solution is described. For \(1<q<2\) and \(p\in [1,\infty)\) it holds NEWLINE\[NEWLINEt^{(1-1/p)/q} \bigl\|u(t,.)-v(t,.) \bigr\|_p\to 0\quad\text{as }t\to \infty,NEWLINE\]NEWLINE where \(v\) is a solution to the problem NEWLINE\[NEWLINE\begin{gathered} v_t+q |v|^{q-1}v_x=0\quad\text{in }(0,\infty) \times \mathbb{R},\\ v(0,.)=M \delta; \end{gathered}NEWLINE\]NEWLINE here \(M=\int u_0(x)dx\). For \(q\geq 2\) and \(p\in [1,\infty)\) it holds NEWLINE\[NEWLINEt^{(1-1/p)/2} \bigl\|u(t,.)-v(t,.) \bigr\|_p \to 0\quad\text{as }t\to \infty,NEWLINE\]NEWLINE where \(v\) is a self-similar fundamental solution to the problem NEWLINE\[NEWLINE\begin{gathered} v_t-\bigl(\widehat a(x)v_x \bigr)_x =0\quad\text{in }(0, \infty)\times\mathbb{R},\\ v(0,.)=M \delta; \end{gathered}NEWLINE\]NEWLINE here \(\widehat a(x)= \alpha\) for \(x<0\) and \(\widehat a(x)=\beta\) for \(x>0\), where \(\alpha,\beta \in\mathbb{R}^+\) are defined by the conditions \(a-\alpha \in W^{1,1} (\mathbb{R}^-)\), \(a-\beta \in W^{1,2} (\mathbb{R}^+)\).