Stability of Geometric Blowup

Abstract

Let u be a solution of (1.1) with initial value u0, smooth for t < t0 and blowing up at time t0 at some pointM0. Our aim is to investigate the stability of this pattern or, more precisely, what can be said about a possible blowup of the solution ū, when ū0 is close enough to u0. Some basic information about this problem can be found, along with further references, in Majda [12], John [10], or [6]. We will work only in local situations, that is, we consider u given in a compact influence domain D bounded below and above by portions  and ω of the planes {t = 0} and {t = t0}, with M0 ∈ ω (and by some lateral surface also). To display solutions u blowing up at M0, we use the concept of “geometric blowup solution” developed in [1]. The reasons for this are the following: i) For n = 1, these solutions are more general than the classical “simple waves” and are known to occur in a number of natural situations [2]. ii) Forn = 2, one-dimensional simple waves are clearly unstable and the geometric blowup solutions are the only examples we know. Moreover, they occur naturally in a number of truly multi-dimensional situations (see, for instance, [3, 4, 5]).