Intermediate asymptotics of the porous medium equation with sign changes. (English) Zbl 1007.35041

The authors study the local behavior of solutions of the porous medium equation \[ u_t =\tfrac{1}{m}\Delta|u|^{m-1}u,\qquad m > 1,\quad (x,t)\in\mathbb{R}^N\times (0,\infty), \quad N\geq 1 \] with sign changes and examine the way as the number of sign changes decreases. Moreover, a formal classification of selfsimilar and nonselfsimilar scenarios for their disappearance are given restricting attention to the radial case for \(N > 1\). For the slow diffusion case, Theorem 1 gives for each \(j = 1,2,3,\dots \) a radial similarity solution of the equation having \(j\) sign changes for \(t < 0\), all of which disappear at \(t = 0\), where \(u(r,0) = Cr^{k_j}\). For \(t > 0\) this solution may be continued as a selfsimilar solution which becomes strictly positive. Odd solutions are also possible, and these are described in Theorem 2. For fast diffusion, i.e., \(0 < m < 1\), the corresponding results are given by Theorem 3. In a sharp contrast to the slow diffusion case, a sign change may also persist for all time. In addition to possible interior extinction of sign changes, Theorems 3,4 give a family of self-similar solutions which describe how any finite number of sign changes may disappear at infinity. Finally, the authors discuss the existence of the so-called tail-switch solution (Theorem 6) and the limit behaviour of the sign-changing similarity solutions.


35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
76S05 Flows in porous media; filtration; seepage