Some qualitative aspects of a nonlinear heat equation with variable coefficients (Q2785615)

Consider the following Cauchy problem: NEWLINE\[NEWLINEu_t= (D(x,t)(u^m)_x)_x -b(x,t)u^p\quad\text{in } \mathbb{R}\times(0,\infty), \qquad u(\cdot,0) = u_0\quad\text{in } \mathbb{R},NEWLINE\]NEWLINE where \(m\) and \(p\) are positive parameters and \(D\) and \(b\) are positive, sufficiently smooth functions. \textit{R. Kersner} and \textit{F. Nicolosi} [J. Math. Anal. Appl. 170, 551-566 (1992; Zbl 0799.35131)] showed that solutions of this problem satisfy two qualitative properties when \(m > 1\) and \(p<1\): the solution becomes identically zero in finite time and the support of \(u(\cdot,t)\) is compact for any \(t > 0\). The purpose of this paper is to extend these results to the parameter range \(0 < p < m \leq1\). When \(D\), \(b\) and \(u_o\) satisfy suitable conditions, the methods of Kersner and Nicolosi are used in the new regime.