Miguel A. Herrero

Boundedness and blow up for a semilinear reaction-diffusion system

Abstract We consider the semilinear parabolic system (S) u t − Δu = ν p ν t − Δν = u q , where x∈RNN ⩾ 1, t > 0, and p, q are positive real numbers. At t = 0, nonnegative, continuous, and bounded initial values (u0, v0(x)) are prescribed. The corresponding Cauchy problem then has a nonnegative classical and bounded solution (u(t, x), v(t,x)) in some strip ST = [0,T) × R N, 0 T∗ = sup {T > 0:u, v remain bounded in S T } . We show in this paper that if 0 T∗ = + ∞ , so that solutions can be continued for all positive times. When pq > 1 and (γ + 1) (pq − 1) ⩾ N 2 with γ = max {p, q}, one has T∗ for every nontrivial solution (u, v). T∗ is then called the blow up time of the solution under consideration. Finally, if (γ + 1)(pq − 1) N 2 both situations coexist, since some nontrivial solutions remain bounded in any strip SΓ while others exhibit finite blow up times.

Singularity patterns in a chemotaxis model

The authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time. Such a finite-time blow-up of solutions implies chemotactic collapse, namely, concentration of species to form sporae. The model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity, which has the form ut=Δu−χ(uv), 0=Δv+(u−1), on ΩR2, where Ω is an open disc with no-flux (homogeneous Neumann) boundary conditions. The initial conditions are continuous functions u(x,0)=u0(x)≥0, v(x,0)=v0(x)≥0 for xΩ. Under these conditions, the authors prove there exists a radially symmetric solution u(r,t) which blows up at r=0, t=T<∞. A specific description of such a solution is presented. The authors also discuss the strong similarity between the chemotactic model they study and the classical Stefan problem.

A semilinear parabolic system in a bounded domain

SummaryConsider the system(S)

Finite-time aggregation into a single point in a reaction - diffusion system

\left\{ \begin{gathered} u_t - \Delta u = v^p , in Q = \{ (t, x), t > 0, x \in \Omega \} , \hfill \\ v_t - \Delta v = u^q , in Q , \hfill \\ u(0, x) = u_0 (x) v(0, x) = v_0 (x) in \Omega , \hfill \\ u(t, x) = v(t, x) = 0 , when t \geqslant 0, x \in \partial \Omega , \hfill \\ \end{gathered} \right.

On the dynamics of social conflicts: looking for the black swan

where Ω is a bounded open domain in ℝN with smooth boundary, p and q are positive parameters, and functions u0 (x), v0(x) are continuous, nonnegative and bounded. It is easy to show that (S) has a nonnegative classical solution defined in some cylinder QT=(0, T) × Ω with T ⩽ ∞. We prove here that solutions are actually unique if pq ⩾ 1, or if one of the initial functions u0, v0 is different from zero when 0 < pq < 1. In this last case, we characterize the whole set of solutions emanating from the initial value (u0, v0)=(0, 0). Every solution exists for all times if 0 1, solutions may be global or blow up in finite time, according to the size of the initial value (u0, v0).

Self-similar blow-up for a reaction-diffusion system

Abstract Consider the Cauchy problem u t − u x x − F ( u ) = 0 ; x ∈ ℝ , t > 0 u ( x , 0 ) = u 0 ( x ) ; x ∈ ℝ where u0(x) is continuous, nonnegative and bounded, and F(u) = up with p > 1, or F(u) = eu. Assume that u blows up at x = 0 and t = T > 0. In this paper we shall describe the various possible asymptotic behaviours of u(x, t) as (x, t) → (0, T). Moreover, we shall show that if u0(x) has a single maximum at x = 0 and is symmetric, u0(x) = u0(−x) for x > 0, there holds 1) If F(u) = up with p > 1, then lim t ↑ T u ( ξ ( ( T − t ) | log ( T − t ) | ) 1 / 2 , t ) × ( T − t ) 1 / ( p − 1 ) = ( p − 1 ) − ( 1 / ( p − 1 ) ) [ 1 + ( p − 1 ) ξ 2 4 p ] − ( 1 / ( p − 1 ) ) uniformly on compact sets |ξ| ≦ R with R > 0, 2) If F(u) = eu, then lim t ↑ T ( u ( ξ ( ( T − t ) | log ( T − t ) | ) 1 / 2 , t ) + log ( T − t ) ) = − log [ 1 + ξ 2 4 ] uniformly on compact sets |ξ| ≦ R with R > 0.

Liouville theorems and blow up behaviour in semilinear reaction diffusion systems

We consider the following system: which has been used as a model for various phenomena, including motion of species by chemotaxis and equilibrium of self-attracting clusters. We show that, in space dimension N = 3, (S) possess radial solutions that blow-up in a finite time. The asymptotic behaviour of such solutions is analysed in detail. In particular, we obtain that the profile of any such solution consists of an imploding, smoothed-out shock wave that collapses into a Dirac mass when the singularity is formed. The differences between this type of behaviour and that known to occur for blowing-up solutions of (S) in the case N = 2 are also discussed.

Blow–Up Profiles in One–Dimensional. Semilinear Parabolic Problems

Consider the initial-boundary value problem for ut = Δu − λuq with λ > 0, 0 < q < 1; the initial data are nonnegative and the boundary data vanish. It is well known that the solution becomes extinct in finite time T, i.e., u(x, t) becomes identically zero for t ⩾ T, where T is some positive number. In this paper we study the profile of x → u(x, t) as t → T.