Similarity stabilizes blow up.
This is an extended version of the author's earlier paper [Journées Équations aux Derivées Partielles, Université de Nantes, Exp. No. 12 (1999; Zbl 1004.35062)]. The considered topic is the equation \[ \psi u_t = -L\left( | u| ^{m-1} u \right), \tag{1} \] where \(\psi(x)\) is positive and \(L\) is a self-adjoint strongly elliptic differential operator of order \(2s\). Application of the similarity transformation \[ u(t,x) = \Phi \bigl( \tau(t) \bigr) v \bigl( \tau(t), x \bigr) \qquad (\Phi > 0) \] with \[ \tau'(t) = \Phi \bigl( \tau(t) \bigr)^{m-1} \] yields the equation \[ \psi v_{\tau} = - L \left( | v| ^{m-1} v\right) - \lambda (\tau) \psi v. \tag{2} \] After defining \(\lambda\) by the condition that \(\int \psi | v| ^{m+1}\) be conserved it is proved that the solution of the rescaled equation (2) tends for \(\tau \to \infty\) to a weak solution of the equation \[ 0 = -L \left( | v| ^{m-1} v \right) - \lambda _{\infty} \psi v \] where \(\lambda_{\infty} = \lim_{\tau \to \infty} \lambda(\tau)\). This generalizes the result of \textit{C. Cortázar}, \textit{M. Del Pino} and \textit{M. Elgueta} [Indiana Univ. Math. J. 47, 541--561 (1998; Zbl 0916.35056)]. In the sequel a special case of (1) with \(L\) of order two is considered. The main theorem claims that, under appropriate assumptions, the solution to (1) blows up in finite time while the solution to the rescaled equation (2) remains bounded.
quasilinear parabolic partial differential equation
scientific article; zbMATH DE number 1957259
scientific article