Singularity Formation in Chemotaxis - A Conjecture of Nagai

Consider the initial boundary value problem for the system (S) ut = uxx− (uvx)x, vt = u− av on an interval [0, 1] for t > 0 where a > 0 with ux(0, t) = ux(1, t) = 0. Suppose u0, v0 are positive constants. The corresponding spatially homogeneous global solution U(t) = u0, V (t) = u0/a+ (v0 − u0/a) exp(−at) is stable in the sense that if (u0, v ′ 0) are positive constants, the corresponding spatially homogeneous solution will be uniformly close to (U(·), V (·)). We consider, in sequence space, an approximate system (S′) which is related to (S) in the following sense: The Fourier transform of (uvx)x is replaced by its finite part. We prove: 1. If u0 > a, then in every neighborhood of (u0, v0) there are (spatially non constant) initial data for which the solution of problem (S′) blows up in finite time in the sense that the solution must leave H1(0, 1)×L2(0, 1) in finite time T . Moreover, the solution components u(·, t), v(·, t) each leave L2(0, 1). 2. If u0 > a, then in every neighborhood of (u0, v0) there are (spatially non constant) initial data for which the solution of problem (S) on (0, 1) × (0, Tmax) must blow up in finite time in the sense that the coefficients of the cosine series for (u, v), become unbounded in the sequence product space `1× `1. A consequence of (2) states that in every neighborhood of (u0, v0), there are solutions of (S′) which, if they are sufficiently regular, will blow up in finite time. (Nagai showed that for the original system such solutions are unstable in the sense that if u0 > a, then in every neighborhood of (u0, u0/a), there are spatially non constant solutions which blow up in finite or infinite time in the sense that the solution must leave H1(0, 1)× L2(0, 1) in finite or infinite time. He conjectured that the blow-up time must be finite.)