ESTIMATES FROM BELOW OF BLOW-UP TIME IN A PARABOLIC SYSTEM WITH GRADIENT TERM

Qualitative properties as blow up, decay bounds and extinction in finite time for solutions to linear and nonlinear parabolic problems have received much attention in the recent literature. We refer to the papers [1], [4], [5] and the references therein. We also mention the results of Galaktionov-Mitidieri-Pohoazev ([3], [13]) on blow-up and global existence for different classes of parabolic problems, including higher order parabolic equations. The blow-up phenomena of solutions to various nonlinear problems, particularly for parabolic systems, have been investigated by different authors. For results in this area, the reader can reference the book [19] and the survey paper


Introduction
Qualitative properties as blow up, decay bounds and extinction in finite time for solutions to linear and nonlinear parabolic problems have received much attention in the recent literature.We refer to the papers [1], [4], [5] and the references therein.
We also mention the results of Galaktionov-Mitidieri-Pohoazev ( [3], [13]) on blow-up and global existence for different classes of parabolic problems, including higher order parabolic equations.
The blow-up phenomena of solutions to various nonlinear problems, particularly for parabolic systems, have been investigated by different authors.For results in this area, the reader can reference the book [19] and the survey paper [1].In the case of a single equation, we refer to [2], [15] and [20] for different results about blow-up phenomena.For other contributions in this field, see for elliptic equations [7] and [19]; for reaction diffusion equations [10] - [12]; for systems [6], [8] and [9].We focus our attention on lower bounds for blow-up time of parabolic systems, which are of a great interest in several practical cases (see, for example, [19] and [20]).
Let us consider the following weakly coupled system where t ⋆ is the blow-up time, Ω is a convex domain in R 3 with the origin inside, whose boundary ∂Ω is sufficiently smooth, p > q > 1, and u 0 (x) and v 0 (x) are nonnegative functions in Ω, satisfying the compatibility conditions on ∂Ω; it follows by the maximum principle that in the interval of existence the solution (u(x, t), v(x, t)) is nonnegative.The paper is organized as follows.In Section 2, we derive a first order differential inequality in terms of an appropriate auxiliary function, by using the Talenti-Sobolev inequality present in [17] and [21], which is valid for nonnegative functions defined in a bounded domain Ω ⊂ R 3 , vanishing on ∂Ω.Then we obtain an explicit lower bound of t ⋆ for a classical solution of system (1); this estimate depends on the geometry of Ω, and on the data p, q, u 0 (x) and v 0 (x).We remark that in a forthcoming paper a numerical resolution algorithm of system (1) will be presented, and some numerical examples will be analyzed to confirm our theoretical results.

Estimate from Below
The aim of this section is to obtain a lower bound of the blow-up time t ⋆ of the solution of (1).To this end we introduce the auxiliary function and we say that (u, v) blows up in W −norm if lim t→t * W (t) = +∞.
Theorem 1.Let W (t) be defined in (2) and (u, v) be a classical solution of (1) which becomes unbounded in W -norm at some finite time t ⋆ .Then an estimate from below for t ⋆ is given by where 0 dx and A is a constant which depends on the data.
Proof.We compute For simplicity we calculate separately U ′ and analogously V ′ .
On the other hand, by using the divergence theorem and the boundary conditions (2. 3), we have We also have To estimate the first term on the right hand in (7) we use the inequality (see Lemma A2 in [16]) with valid for any nonnegative C 1 −function ω(x) defined in a bounded domain Ω ⊂ R 3 star-shaped and convex in two orthogonal directions, with n ≥ 1.If n = 2p − 1 and ω = u, we obtain By using the arithmetic inequality and Hölder and Schwarz inequalities respectively in first and second term in (9), we have Successively, in the last term of the right hand of (11) we use both Hölder and the arithmetic inequalities a r b s ≤ ra + sb, r Let us estimate now Ω v 3p dx in (7).By using (8) with n = 2p, ω = v we obtain Now we replace ( 12) and ( 13) in ( 7) to obtain In order to estimate the third term in (4), we observe that with ξ = q 2p+q−1 , and by using (2.10) in [14] we obtain and λ 1 the first eigenvalue of the problem ∆w + λw = 0 x ∈ Ω, w > 0, Hölder inequality allows us to write Replacing ( 18) in ( 16) we get Finally we substitute ( 5), ( 14) and ( 19) in ( 4) and obtain .