A very singular solution for the dual porous medium equation and the asymptotic behaviour of general solutions.

*(English)*Zbl 0756.35038We study the nonlinear parabolic equation (E) \(z_ t=| z_{xx}|^{m-1}z_{xx}\) with \(m>0\). Equations of this form have been proposed in nonlinear elasticity. Our contribution consists in establishing the existence and properties of certain solutions of selfsimilar type which are then used to obtain information on the asymptotic behaviour of general solutions with bounded and compactly supported initial data. In the case \(m=1\), i.e. the classical heat equation, such solutions are the so-called fundamental solutions and a complete theory can be based on representation of general solutions in terms of them.

In this paper we show that for \(m>1\) equation (E) admits nonnegative selfsimilar solutions \(\tilde z(x,t)\) of the form \(\tilde z=t^{- \gamma}F(xt^{-\delta})\) with \(\gamma,\delta>0\), with fast decay as \(x\to\infty\). The exponents depend on \(m\). Actually, they satisfy the relation \(\gamma(m-1)+2m\delta=1\). However, and contrary to what happens in other popular equations, like the heat equation, the porous medium equation \((u_ t=(| u|^{m-1}u)_{xx})\), or the \(p\)-Laplacian equation \((u_ t=(| u_ x|^{p-2}u_ x)_ x)\), there is no second simple relation to determine the coefficients in terms of \(m\), namely they are anomalous exponents. In the paper we describe the precise dependence of \(\gamma\) and \(\delta\) on \(m\).

Though our solutions share some properties of fundamental solutions of the heat equation, they have a strikingly different behaviour as \(t\to 0\): while the former tend to a point mass as time goes to 0 (source-type solutions), our solutions are of a very singular tpe if \(m>1\) in the sense that the initial mass is infinite. They have zero initial mass if \(0<m<1\).

In this paper we show that for \(m>1\) equation (E) admits nonnegative selfsimilar solutions \(\tilde z(x,t)\) of the form \(\tilde z=t^{- \gamma}F(xt^{-\delta})\) with \(\gamma,\delta>0\), with fast decay as \(x\to\infty\). The exponents depend on \(m\). Actually, they satisfy the relation \(\gamma(m-1)+2m\delta=1\). However, and contrary to what happens in other popular equations, like the heat equation, the porous medium equation \((u_ t=(| u|^{m-1}u)_{xx})\), or the \(p\)-Laplacian equation \((u_ t=(| u_ x|^{p-2}u_ x)_ x)\), there is no second simple relation to determine the coefficients in terms of \(m\), namely they are anomalous exponents. In the paper we describe the precise dependence of \(\gamma\) and \(\delta\) on \(m\).

Though our solutions share some properties of fundamental solutions of the heat equation, they have a strikingly different behaviour as \(t\to 0\): while the former tend to a point mass as time goes to 0 (source-type solutions), our solutions are of a very singular tpe if \(m>1\) in the sense that the initial mass is infinite. They have zero initial mass if \(0<m<1\).

Reviewer: F.Bernis