No abstract

This paper dealswith a parabolic–parabolic Keller–Segel-type systemin a bounded domain ofRN, fN D 2; 3g, under different\ud boundary conditions, with time-dependent coefficients and a positive source term. The solutions may blowup in finite\ud time t; and under appropriate assumptions on data, explicit lower bounds for blow-up time are obtained when blow up\ud occur

A Rayleigh-Faber-Krahn type inequality is used to derive bounds for boundary value problems appearing in reaction-diffusion problems where the reactant is consumed. Interesting quantities are the minimum of the solution and the measure of the set where it vanishes. The proofs are rather elementary and apply to problems possessing solutions in a weak sense. Mathematics Subject Classification (2000). 49K20, 35J28, 35J60.

SUMMARYThis paper deals with a class of semilinear parabolic problems. In particular, we establish conditions on the data su cient to guarantee blow up of solution at some ÿnite time, as well as conditions which will insure that the solution exists for all time with exponential decay of the solution and its spatial derivatives. In the case of global existence, we also investigate the continuous dependence of the solution with respect to some data of the problem.

We consider the following quasilinear Keller-Segel system { u t = Δu − ∇(u∇v) + g(u), (x, t) ∈ Ω × [0, T max), 0 = Δv − v + u, (x, t) ∈ Ω × [0, T max), on a ball Ω ≡ B R (0) ⊂ R n , n ≥ 3, R > 0, under homogeneous Neumann boundary conditions and nonnegative initial data. The source term g(u) is superlinear and of logistic type, that is, g(u) = u − u k , k > 1, > 0, > 0, and T max is the blow-up time. The solution (u, v) may or may not blow-up in finite time. Under suitable conditions on data, we prove that the function u, which blows up in L ∞ (Ω)-norm, blows up also in L p (Ω)-norm for some p > 1. Moreover, a lower bound of the lifespan (or blow-up time when it is finite) T max is derived. In addition, if Ω ⊂ R 3 a lower bound of T max is explicitly computable.