Singularity Formation in Chemotaxis--- A Conjecture of Nagai

Consider the initial-boundary value problem for the system (S)u_t = u_xx - (uv_x)_x, v_t= u- av on an interval [0,1] for t > 0, where a > 0 with u_x(0,t) = u_x(1,t)= 0. Suppose \mu, v₀ are positive constants. The corresponding spatially homogeneous global solution U(t) = \mu, V(t) = \mu a + (v₀ - \mu a)\exp(-at) is stable in the sense that if (\mu',v_0' ) are positive constants, the corresponding spatially homogeneous solution will be uniformly close to (U(\cdot),V(\cdot)).

We consider, in sequence space, an approximate system (S') which is related to (S) in the following sense: The chemotactic term (uv_x)_x is replaced by the inverse Fourier transform of the finite part of the convolution integral for the Fourier transform of (uv_x)_x. (Here the finite part of the convolution on the line at a point x of two functions, f,g, is defined as $\int_0^x(f(y)g(y-x)\,dy$.) We prove the following: If \mu > a, then in every neighborhood of (\mu,v₀ ) there are (spatially nonconstant) initial data for which the solution of problem (S') blows up in finite time in the sense that the solution must leave L² (0,1)\times H¹ (0,1) in finite time T. Moreover, the solution components u(\cdot,t),v(\cdot,t) each leave L² (0,1).If \mu > a, then in every neighborhood of (\mu,v₀ ) there are (spatially nonconstant) initial data for which the solution of problem (S) on (0,1) \times (0,T_max ) 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 $\ell^1\times\ell^1_1$.

A consequence of (2) states that in every neighborhood of (\mu,v₀ ), there are solutions of (S) which, if they are sufficiently regular, will blow up in finite time. (Nagai and Nakaki [Nonlinear Anal., 58 (2004), pp. 657--681] showed that for the original system such solutions are unstable in the sense that if \mu > a, then in every neighborhood of (\mu,\mu a), there are spatially nonconstant solutions which blow up in finite or infinite time. They conjectured that the blow-up time must be finite.) Using a recent regularity result of Nagai and Nakaki, we prove this conjecture.