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Reaction-diffusion problems under non-local boundary conditions with blow-up solutions
Journal of Inequalities and Applications volume 2014, Article number: 167 (2014)
Abstract
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.
MSC:35K55, 35K60.
1 Introduction
Quittner and Souplet in [1] 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 [4] 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 [2] and [10–12] for systems.
In this paper we consider a class of reaction-diffusion equations where a space integral is present on the boundary condition and time dependent coefficients are present both in the nonlinearity term and in the boundary condition. Our aim is to introduce conditions on the data and geometry of the spatial domain, sufficient for the solution to blow up in finite time . Moreover, lower and upper bounds are derived. More precisely we consider the following problem:
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 [12] and differential inequality techniques.
If , (2) becomes the usual Neumann boundary condition and we obtain the Payne-Philippin’s result contained in [5], 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.
Assume that the functions f and g satisfy
where
and
Define
If the solution becomes unbounded in Φ-measure at time , then
where T is implicitly given as
where and are two suitable positive functions.
Theorem 1.2 Let be a (non-negative) classical solution of (1)-(3) with Ω a bounded domain in . Assume that the functions f and g satisfy
Moreover, assume
with
Then no solution can exist for all time, but it blows up in and hence in norm, , at time , with and T implicitly defined by
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.
For physical motivation of such problems we refer the reader to [13–16] and the references therein.
2 Lower bounds
First we state an inequality that plays a basic role in the proofs of our results.
Lemma 2.1 Let W be any non-negative function and Ω a bounded convex domain in , , with the origin inside. Then, for any , the following inequality holds:
where
Proof We start from the following Sobolev type inequality derived in [8] and [9]:
By applying the Hölder inequality, the first term on the right of (16) becomes
Moreover, by applying the Schwarz and Hölder inequalities to the second term on the right of (16) we get
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.
For brevity, we let and , . We recall that from (5) we have . We compute
By applying the divergence theorem, boundary conditions (2) and (4), we have
For convenience, set ; we get
We now replace (20), (21) in (19) and use (6) to obtain
To estimate the second term on the right-hand side of (22) we use the inequality (3.8) in [5], i.e.
where γ is an arbitrary positive constant to be chosen later.
Now we estimate the second term in (23) by using (7) and the inequalities (48), (51) in [5], valid in a convex domain . We obtain
with , , and d in (15) with , and is an arbitrary constant.
Note that deriving the second and third inequality in (24), we make use of the inequality
and of the arithmetic inequality
Inserting (23) and (24) in (22) we get
where .
We now estimate separately the two factors in the last term in (27). For the first, making use of Lemma 2.1 with and , we have
In the second, by using the Hölder inequality and hypothesis (5), we obtain
By using (28) and (29), we get
where in the last inequality we use (26) with arbitrary.
We now replace (30) in (27) to have
with
We now choose γ, σ, ϵ positive constants such that .
A possible choice of γ, σ, ϵ, is
Then
Note that A, B, C are positive constants, whereas D and E may depend on the time through the coefficients and .
In order to simplify (31), set
and we obtain and . Now (31) can be rewritten (with ) as
In the second term in (35) we now use the Schwarz inequality to obtain
Moreover, since , we write
with , and an arbitrary positive constant.
Analogously
where we again use (26) with , and an arbitrary positive constant.
Moreover, we insert (36), (37), and (38) in (35) so that the differential inequality (35) can be rewritten as
with
From (39) we can write
and we set to obtain
Then define , and (41) may be rewritten as
From this we obtain
i.e.
Then if , we have
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.
In this case we have
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.
To this end we compute
where in the last inequality we used (10).
For any , by the Hölder inequality
we have
By using (44) in (43) we get
Now we observe that the function Ψ is non-decreasing, then .
Since ,
i.e.
By inserting (46) in (45), we have
with defined in (11).
Now we integrate (47) from 0 to t and we obtain the inequality
Using (11) we see that the inequality (48) cannot hold for all time, but u will blow up in norm (hence in norm, ) at a finite time and
Moreover, let . Since k is increasing, there exists the inverse function and we can write
which is the desired upper bound of .
Another upper bound for can be obtained by means of a new auxiliary function so defined:
with satisfies
Under this condition no (non-negative) solution of problem (1)-(3) can exist for all time, but it blows up in χ norm at time , with and T implicitly defined by
In fact, following the proof of Theorem 1.2, we obtain the differential inequality
where .
Integrating (53) over , we obtain
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).
Remark The result can be extended to the case . In fact, by (53) we get
By assuming that
we conclude that the solution blows up in χ norm at time , with and T implicitly defined by
Authors’ information
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).
References
Quittner P, Souplet P Birkhäuser Advanced Texts. In Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States. Birkhäuser, Basel; 2007.
Marras M, Vernier Piro S: Blow-up phenomena in reaction-diffusion systems. Discrete Contin. Dyn. Syst. 2012,32(11):4001-4014.
Marras M, Vernier Piro S: On global existence and bounds for blow-up time in non-linear parabolic problems with time dependent coefficients. Discrete Contin. Dyn. Syst. 2013, 2013: 535-544. suppl.
Song JC: Lower bounds for the blow-up time in a non-local reaction-diffusion problem. Appl. Math. Lett. 2011, 24: 793-796. 10.1016/j.aml.2010.12.042
Payne LE, Philippin GA: Blow up phenomena in parabolic problems with time dependent coefficients under Neumann boundary conditions. Proc. R. Soc. Edinb. 2012, 142: 625-631. 10.1017/S0308210511000485
Payne LE, Philippin GA: Blow up in a class of non-linear parabolic problems with time dependent coefficients under Robin type boundary conditions. Appl. Anal. 2012, 91: 2245-2256. 10.1080/00036811.2011.598865
Payne LE, Philippin GA, Vernier Piro S: Blow up phenomena for a semilinear heat equation with nonlinear boundary condition, I. Z. Angew. Math. Phys. 2010, 61: 999-1007. 10.1007/s00033-010-0071-6
Payne LE, Philippin GA, Vernier Piro S: Blow up phenomena for a semilinear heat equation with nonlinear boundary condition, II. Nonlinear Anal. 2010, 73: 971-978. 10.1016/j.na.2010.04.023
Payne LE, Schaefer PW: Lower bounds for blow-up time in parabolic problems under Neumann boundary conditions. Appl. Anal. 2006, 85: 1301-1311.
Marras M: Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions. Numer. Funct. Anal. Optim. 2011, 32: 453-468. 10.1080/01630563.2011.554949
Marras M, Vernier Piro S: Bounds for blow-up time in nonlinear parabolic system. Discrete Contin. Dyn. Syst. 2011, 2011: 1025-1031. suppl.
Payne LE, Schaefer PW: Blow-up phenomena for some nonlinear parabolic systems. Int. J. Pure Appl. Math. 2008, 42: 193-202.
Hu B, Yin HM: Semilinear parabolic equations with prescribed energy. Rend. Circ. Mat. Palermo 1995, 44: 479-505. 10.1007/BF02844682
Rubinstein J, Sternberg P: Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 1992, 48: 249-264. 10.1093/imamat/48.3.249
Marras M, Vernier Piro S: Blow up and decay bounds in quasilinear parabolic problems. Discrete Contin. Dyn. Syst. 2007, 2007: 704-712. suppl.
Payne LE, Philippin GA: Blow up phenomena for a class of parabolic systems with time dependent coefficients. Appl. Math. 2012, 3: 325-330. 10.4236/am.2012.34049
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Marras, M., Vernier Piro, S. Reaction-diffusion problems under non-local boundary conditions with blow-up solutions. J Inequal Appl 2014, 167 (2014). https://doi.org/10.1186/1029-242X-2014-167
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DOI: https://doi.org/10.1186/1029-242X-2014-167