Verlag der Österreichischen Akademie der Wissenschaften

Abstract:
We study finite difference schemes for axisymmetric blow-up solutions ofa nonlinear heat equation in higher spatial dimensions. The phenomenology of blow-up in higher-dimensional space is much more complex than that in one space dimension.To obtain a more complete picture for such phenomena from computational results, it is useful to know the technical details of the numerical schemes for higher spatial dimensions.Since first-order differentiation appears in the differential equation,we pay special attention to it. A sufficient condition for stabilityis derived. In addition to the convergence of the numerical blow-up time,certain blow-up behaviors, such as blow-up sets and blow-up in the $L^p$-norm,are taken into consideration. It is sometimes experienced that a certain property of solutions of a partial differential equation maybe lost by a faithfully constructed convergent numerical scheme. The phenomenon of one-point blow-up is a typical example in thenumerical analysis of blow-up problems. We prove that our scheme can preserve such a property.It is also remarkable that the $L^p$-norm $(1\leq p<\infty)$ of the solution of the nonlinear heat equation mayblow up simultaneously with the $L^\infty$-norm or remains bounded in $[0,T)$, where $T$ denotes the blow-uptime of the $L^\infty$-norm. We propose a systematic way to compute numerical evidence of the $L^p$-norm blow-up.The computational results are also analyzed. Moreover, we prove an abstract theorem which shows the relationshipbetween the numerical $L^p$-norm blow-up and the exact $L^p$-norm blow-up. Numerical examplesfor higher-dimensional blow-upsolutions are presented and discussed.