Reaction-diffusion problems under non-local boundary conditions with blow-up solutions
© Marras and Vernier Piro; licensee Springer. 2014
Received: 28 October 2013
Accepted: 9 April 2014
Published: 6 May 2014
This paper deals with blow-up solutions to a class of reaction-diffusion equations under non-local boundary conditions. We prove that under certain conditions on the data the blow-up will occur at some finite time and when the blow-up does occur, lower and upper bounds are derived.
Quittner and Souplet in  consider different classes of reaction-diffusion problems with non-local source terms involving space integrals and investigate under what conditions the solutions blow up or exist globally (see also [2, 3]). Recently Song  has considered parabolic problems under Dirichlet or Neumann boundary conditions, containing a non-local term in the nonlinearities and, for solutions that blow up at some finite time, they derive lower bounds for the blow-up time. For other contributions in this field, see [3, 5, 6] and [7–9] for reaction-diffusion equations, and see  and [10–12] for systems.
where Ω is a bounded domain in , with smooth boundary, is the outward normal derivative of u on the boundary ∂ Ω, , , are smooth non-negative functions, is the blow-up time if blow-up occurs, and the time dependent coefficients , , are positive and regular functions. Moreover, satisfies the compatibility condition on the boundary. Note that for the maximum principle. The results are based on some Sobolev type inequalities  and differential inequality techniques.
If , (2) becomes the usual Neumann boundary condition and we obtain the Payne-Philippin’s result contained in , Theorem 2.
Now we state the main theorems of this paper.
Theorem 1.1 Let be a (non-negative) classical solution of problem (1)-(3) with Ω a bounded convex domain in with the origin inside.
where and are two suitable positive functions.
The paper is organized as follows. In Section 2 we obtain a lower bound for under the hypothesis of convexity of Ω and suitable conditions on data and time dependent coefficients.
In Section 3, we consider the problem (1)-(3) under conditions on the data which ensure that no solution can exist for all time. In fact the solution blows up at some finite time in and hence in norm () and upper bounds for are derived. We note that we obtain blow-up, even if the coefficients are constants and also for functions decreasing not too fast at infinity.
2 Lower bounds
First we state an inequality that plays a basic role in the proofs of our results.
By inserting (17) and (18) in (16), we get (14).
Our aim is now to derive a lower bound for .
We assume that becomes unbounded at some finite time , and under the conditions of Theorem 1.1, we derive a lower bound for , which works for values of not too small.
where γ is an arbitrary positive constant to be chosen later.
with , , and d in (15) with , and is an arbitrary constant.
where in the last inequality we use (26) with arbitrary.
We now choose γ, σ, ϵ positive constants such that .
Note that A, B, C are positive constants, whereas D and E may depend on the time through the coefficients and .
with , and an arbitrary positive constant.
where we again use (26) with , and an arbitrary positive constant.
Clearly T is implicitly given by (9). □
Remark 1 Note that the bound (9) is a good estimate of because we consider our problem (1)-(3) with initial data not too small. For instance we can choose such that .
Remark 2 If and are constants then and are constants.
3 Blow-up of u in finite time and upper bounds. Proof of Theorem 1.2
In this section we establish that under the hypotheses of Theorem 1.2, no solution can exist for all time, but it blows up in and hence in norm, , at time . Then an upper bound of is obtained.
where in the last inequality we used (10).
Now we observe that the function Ψ is non-decreasing, then .
with defined in (11).
which is the desired upper bound of .
We see that the inequality (54) cannot hold for all time, but u will blow up in χ norm at a finite time . At the end we get the upper bound T, with T implicitly defined by (52).
The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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