Volume 2010, Article ID 491023,11pages doi:10.1155/2010/491023
Research Article
Large Solutions for Semilinear Parabolic Equations Involving Some Special Classes of Nonlinearities
Constantin P. Niculescu and Ionel Rovent¸a
Department of Mathematics, University of Craiova, 200585 Craiova, Romania
Correspondence should be addressed to Ionel Rovent¸a,[email protected] Received 8 April 2010; Accepted 21 June 2010
Academic Editor: Yong Zhou
Copyrightq2010 C. P. Niculescu and I. Rovent¸a. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a new class of nonlinearities for which a nonlocal parabolic equation with Neumann boundary conditions has finite time blow-up solutions. Our approach is inspired by previous work done by Jazar and Kiwan2008and El Soufi et al.2007.
1. Introduction
This paper is devoted to the existence of large solutions of the semilinear parabolic problem
ut−Δuf|u|− 1 mΩ
Ωf|u|dx, inΩ,
∂u
∂n 0, on∂Ω,
1.1
with the initial conditions
ux,0 u0x, onΩ, where
Ωu0dx0. 1.2
Here Ω ⊂ RN is a bounded regular domain of class C2, f : 0,∞ → 0,∞ is a locally Lipschitz function, mΩrepresents the Lebesgue measure of the domain Ω,and Δis the Laplace operator.
The above problem was recently studied by El Soufi et al.1and Jazar and Kiwan2, under the assumption thatfis a power function of the formfu upwithp >1.Under the
same restriction onf, some lower bounds estimates for the blow-up time were established in 3. See also4,5.
The aim of our paper is to extend their results to a larger class of nonlinearities whose precise definition is as follows.
Definition 1.1. A real-valued functionf defined on an intervala,∞ witha ≥ 0satisfies propertyCif it is locally Lipschitz, nonnegative, and its mean value1/t−at
afxdx
has a superlinear growth in the sense that the ratio 1/t−at
afxdx
t−aα 1.3
is nondecreasing fortlarge enough and someα >1.
The monotonicity condition on1.3means precisely the existence of a constantC ∈ 0,1/2 precisely,C 1 α−1such that
Cft≥ 1 t−a
t
a
fxdx 1.4
fort > alarge enough.
For example, ifg ∈ C10,∞,g0 0,andg is nondecreasing, then the function ft gttα, withα >1, satisfies propertyC. In fact,
t
0
fxdx tα 1 α 1gt−
t
0
gxxα 1 α 1dx
≤ tα 1
α 1gt Ctft,
1.5
whereC1/α 1∈0,1/2.
Assuming thatf0 0which is the case ifa0 and1.4works for allt > 0, one can infer from1.4that
ft f0
2 ≥ 1
t t
0
fxdx, 1.6
a fact that reminds of the Hermite-Hadamard inequality in convex functions theory. See6, page 50. Thus propertyCcan be ascribed to the field of generalized convexity.
The problems of type 1.1 and 1.2 arise naturally in mechanics, biology, and population dynamics. For example, if we consider a couple or a mixture of two equations of the above type, the resulting problem describes the temperatures of two substances, which constitute a combustible mixture, or represents a model for the behavior of densities of two diffusion biological species which interact with each other. This type of problems is connected also with parabolic systems of heat equations with local sources, which arise in population dynamics. See4,7–11.
Our paper is organized as follows. InSection 2we show that every solutionuof the problems1.1and 1.2 withu0 not identically 0 andf satisfying propertyCis large, provided that its energy att 0 is nonpositive. SeeTheorem 2.4. Our approach combines previous work done by El Soufi et al.1, with a careful analysis of the properties of energy of solutions.
InSection 3we discuss the connection of property Cwith other special classes of nonlinearities, well known in the literature. We prove that every function with generalized regular variation`a la Karamata, as well as everyN-function in the sense of Orlicz, satisfies property C.Meantime property Cand the classical Keller-Osserman condition have a large overlapthough they are distinct from each other. Thus the class of functions satisfying propertyCprovides indeed a natural framework for the existence of large solutions for the problems1.1and1.2.
2. The Existence of Large Solutions
The existence of a solution to the problems 1.1 and 1.2 can be found in 1. It can be summarized as follows.
Theorem 2.1. Assume thatΩ⊂RNis a bounded regular domain of classC2andf:0,∞→0,∞ is a locally Lipschitz function. Then for everyu0 ∈CΩthere is an elementtmax∈0,∞such that the problems1.1and1.2has a unique solution
u∈C
0, tmax;C Ω
∩C1
0, tmax;C Ω
, 2.1
which solves the integral equation
ut etδu0 t
0
et−sΔfusds 2.2
on0, tmax. Moreover,
Ωutdx0, ∀t∈0, tmax, 2.3 and iftmax<∞,then limt→tmaxutL∞Ω∞.
Each solutionuof the problems1.1and1.2has the property
Ωu dx 0 because the integral in the right-hand side of1.1is 0 and
d dt
Ωu dx
Ωutdx
ΩΔu dx0. 2.4 Hence, by the initial condition1.2, we have
Ωu dx0.
Lemma 2.2. Letu∈CΩbe a solution of 1.1and1.2. Then the energy ofuat the momentt,
Et
Ω
1 2|∇u|2−
u
0
f|t|dt
dx, 2.5
verifies the formula
Et E0− t
0
Ωu2tdx dt, ∀t >0. 2.6 Proof. In fact,
dE dtt
Ω
∇ut∇u−utf|u| dx
∂Ω
∂u
∂nutdσ−
ΩutΔu dx−
Ωutf|u|dx −
Ωut
Δu f|u| dx−
Ωu2tdx,
2.7
and by integrating both sides over0, t, we obtain formula2.6.
According to the previous lemma, ifE0is nonpositive, thenEtis nonpositive for allt >0.In the case of functionsfsatisfying conditionC,this leads to
C
Ωuf|u|dx≥
Ω
u
0
f|t|dt dx≥ 1 2
Ω|∇u|2. 2.8
Lemma 2.3. Under the assumptions ofLemma 2.2consider the two auxiliary functions
mt: 1 2
Ωu2x, tdx, ht: t
0
msds. 2.9
Then
mt≥ 1 C
t
0
Ωu2tdt, 2.10
mt≥ 1
2C−1
λmt, for some λ >0, 2.11
1 2C
ht−h0 2 ≤htht, 2.12
provided thatfsatisfies conditionC.
Proof. In fact,
mt
Ωutu dx
Ωu
Δu f|u| dx
≥
Ω
−|∇u|2 1 C
u
0
f|t|dt
dx
−1 C
Ω
1 2|∇u|2−
u
0
f|t|dt
dx 1
2C−1
Ω|∇u|2dx.
2.13
Hence,
mt≥ −1 CEu
1 2C−1
Ω|∇u|2dx
≥ −1
CEu −1
CEu0 1 C
t
0
Ωu2t dx dt
≥ 1 C
t
0
Ωu2tdx dt.
2.14
On the other hand, by the Poincar´e inequality, we have
mt≥ 1
2C−1
Ω|∇u|2dx≥ 1
2C−1
λ
Ωu2dx 1
2C−1
λmt, 2.15
whereλis a suitable positive constant.
We pass now to the proof of2.12. Since
ht−h0 t
0
msds t
0
Ωuutdx dt
≤ t
0
Ωu2dx dt
1/2t 0
Ωu2tdx dt 1/2
≤2ht1/2
Cmt 1/2
2Chtht 1/2,
2.16
by2.10we infer that
ht−h0 t
0
msds≥0, 2.17
and thus
1 2C
ht−h0 2≤htht. 2.18
We are now in a position to state the main result of our paper.
Theorem 2.4. Assume thatf : 0,∞ → 0,∞is a function with propertyC,and letube the solution of the problems1.1and1.2corresponding to an initial datau0∈CΩ,u0not identically zero. If the energy ofuatt0 is nonpositive, thenu,as a function oft,cannot be inL∞0, T;L2Ω for allT >0.
Proof. Suppose, by reduction ad absurdum, that the solutionux,·exists in
L∞
0, T;L2Ω
2.19
for allT >0. By2.11,
t→ ∞limht lim
t→ ∞mt ∞, 2.20
which yields, for eachβ∈0,1/C,the existence of a numberT0 >0 such that for allt > T0,
βht2≤ 1 C
ht−h0 2. 2.21
Now, by2.12we obtain
βht2≤2htht. 2.22
We will show, by considering the functionHt ht−q, for a suitableq >0,that the last inequality leads to a contradiction. In fact,
Ht qht−q−2
q 1 ht 2−htht
≤qht−q−2 2
q 1 β −1
htht,
2.23
for allt ≥ T0,so that forβ ∈ 0,1/Candq ∈ 0,1/2C−1with 2q 1 < β < 1/C,the corresponding functionHtis concave.
By2.20, limt→ ∞ht ∞, whence limt→ ∞Ht 0. ThusHprovides an example of a concave and strictly positive function which tends to 0 at infinity, a fact which is not possible.
The proof is done.
3. Classes of Functions with Property C
The aim of this section is to comment on how large is the class of functions which plays propertyC. In this respect we will discuss here several particular classes of functions with this property.
We start with the class of regularly varying functions, introduced by Karamata in12.
Definition 3.1. A positive measurable functionf defined on intervala,∞ with a ≥ 0is said to be regularly varying at infinity, of indexσ∈Rabbreviated,f ∈RV∞σ, provided that
xlim→ ∞
ftx
fx tσ, ∀t >0. 3.1
All functions of indexσare of the form fx xσexp
ax
x
0
εs s ds
, 3.2
whereaxandεxare bounded and measurable,ax → α∈R, andεx → 0 asx → ∞.
In particular, so are
xσlogx, xσloglogx, xσexp
logx loglogx
, xσexp
logx 1/3 cos
logx 1/3 . 3.3 See13for details.
Semilinear problems with nonlinearities in the class of regularly varying functions have been studied by Cˆırstea and R˘adulescu14.
Proposition 3.2. Iff ∈RV∞σwithσ >1, then
xlim→ ∞
Fx
xfx 1
σ 1 < 1
2, 3.4
where
Fx: x
0
fsds. 3.5
Under these assumptions,fsatisfies conditionC(and thusTheorem 2.4applies to it).
Proof. To prove this, consider the change of variablestx,which yields
Fx x
0
fsds
1
0
xftxdt. 3.6
The continuity offand the fact thatf ∈RV∞σassure the existence of aδ >0 such that for everyx > δwe have
ftx
fx ≤tσ 1, 3.7
whence the integrability of the functiont → ftx
fx on0,1. Then
xlim→ ∞
Fx xfx lim
x→ ∞
1
0
ftx fxdt
1
0 xlim→ ∞
ftx fxdt
1
0
tσdt 1 σ 1,
3.8
where the commutation of the limit with the integral is motivated by the Lebesgue dominated convergence theorem.
An important class of nonlinearities which appeared in connection with the study of boundary blow-up problems for elliptic equations is that of functions satisfying the Keller- Osserman condition. See the papers by R˘adulescu15and Dumont et al.16.
Definition 3.3. A nonnegative and nondecreasing function f ∈ C10,∞ with f0 0 satisfies the generalized Keller-Osserman condition of orderp >1 if
∞
1
1
Ft1/pdt <∞, 3.9
whereFis the primitive offgiven by formula3.5.
If f ∈ RV∞σ 1 with σ 2 > p > 1 a nondecreasing and continuous function, then F ∈ RV∞σ 2 andF−1/p ∈ RV∞−σ−2/p. Since−σ−2/p < −1, we infer that F−1/p∈L11,∞and thusfsatisfies the generalized Keller-Osserman condition.
It is worth to notice that the function expt is not regularly varying at infinity though satisfies the generalized Keller-Osserman condition and also the hypothesis of Proposition 3.4.
Necessarily, if a functionfsatisfies the generalized Keller-Osserman condition of order p >1, then
t→ ∞lim Ft
tp ∞, 3.10
whileFt/tpmay beor may be nota monotonic function.
As noticed in the Introduction, propertyCis intimately related to the monotonicity ofFt/tpin the following way.
Proposition 3.4. IfFt/tpis nondecreasing for somep >2, then the functionfsatisfies condition CwithC1/p(and thusTheorem 2.4applies to it).
According toProposition 3.4, the functionft ptp−1logt 1 tp/t 1satisfies for p >2 conditionC but not the generalized Keller-Osserman condition of orderp.Indeed,f admits the primitiveFt tplogt 1.
We end our paper by discussing the connection of property C with a class of functions due to Orlicz.
Definition 3.5. AnN-function is any functionM:0,∞ → Rof the form
Mx x
0
ptdt, 3.11
wherepis nondecreasing and right continuous,p0 0,pt>0 fort >0,and limt→ ∞pt
∞.
AnN-functionMsatisfies theΔ2-condition if there exist constantsk > 0 andx0 ≥ 0 such that
M2x≤kMx, ∀x≥x0. 3.12
AnyN-functionMis convex and plays the following properties:
N1M0 0 andMx>0 forx >0;
N2Mx/x → 0 asx → 0 andMx/x → ∞asx → ∞.
Two examples ofN-functions which satisfy theΔ2-condition arexp/pforp≥1and tlogt .
The N-functions which satisfy the Δ2-condition are instrumental in the theory of Orlicz spaceswhich extend theLpμspaces. Their theory is available in many books, such as17,18, and has important applications to interpolation theory19and Fourier analysis 20.
According to18, page 23, the constantk which appears in the formulation ofΔ2- condition is always greater than or equal to 2.
Proposition 3.6. EveryN-functionM:0,∞ → Rwhich satisfies theΔ2-condition has property C(and thusTheorem 2.4applies to it).
Proof. SinceMis nondecreasing,
Mtx M
2log2tx
≤M
2log2t 1x
, 3.13
and taking into account theΔ2-condition we infer that
Mtx≤Mxklog2t 1≤Mxklog2t 1≤Mxt2log2k, 3.14
for allxbig enough andt≥2.Hence, t
0
Mxdx
1
0
tMtsds
≤ 1
0
tMts2log2kds 1
2log2k 1tMt
≤ 1 3tMt,
3.15
and the proof is done.
Acknowledgment
The authors have been supported by CNCSIS Grant 420/2008.
References
1 A. El Soufi, M. Jazar, and R. Monneau, “A gamma-convergence argument for the blow-up of a non- local semilinear parabolic equation with Neumann boundary conditions,” Annales de l’Institut Henri Poincar´e. Analyse Non Lin´eaire, vol. 24, no. 1, pp. 17–39, 2007.
2 M. Jazar and R. Kiwan, “Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions,” Annales de l’Institut Henri Poincar´e. Analyse Non Lin´eaire, vol. 25, no. 2, pp.
215–218, 2008.
3 L. E. Payne and P. W. Schaefer, “Lower bounds for blow-up time in parabolic problems under Dirichlet conditions,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1196–1205, 2007.
4 L. A. Caffarelli and A. Friedman, “Blowup of solutions of nonlinear heat equations,” Journal of Mathematical Analysis and Applications, vol. 129, no. 2, pp. 409–419, 1988.
5 F. Gladiali and G. Porru, “Estimates for explosive solutions top-Laplace equations,” in Progress in Partial Differential Equations, Vol. 1 (Pont-`a-Mousson, 1997), vol. 383 of Pitman Research Notes in Mathematics Series, pp. 117–127, Longman, Harlow, UK, 1998.
6 C. P. Niculescu and L.-E. Persson, Convex Functions and Their Applications. A Contemporary Approach, vol. 23 of CMS Books in Mathematics, Springer, New York, NY, USA, 2006.
7 J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, vol. 83 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
8 R. S. Cantrell and C. Cosner, “Diffusive logistic equations with indefinite weights: population models in disrupted environments. II,” SIAM Journal on Mathematical Analysis, vol. 22, no. 4, pp. 1043–1064, 1991.
9 Y. Chen, “Blow-up for a system of heat equations with nonlocal sources and absorptions,” Computers
& Mathematics with Applications, vol. 48, no. 3-4, pp. 361–372, 2004.
10 J. Furter and M. Grinfeld, “Local vs. nonlocal interactions in population dynamics,” Journal of Mathematical Biology, vol. 27, no. 1, pp. 65–80, 1989.
11 G. Izzo, Y. Muroya, and A. Vecchio, “A general discrete time model of population dynamics in the presence of an infection,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 143019, 15 pages, 2009.
12 J. Karamata, “Sur un mode de croissance r´eguli`ere des fonctions,” Mathematica, Cluj, vol. 4, pp. 38–53, 1930.
13 N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, vol. 27 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, Mass, USA, 1989.
14 F. C. Cˆırstea and V. R˘adulescu, “Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach,” Asymptotic Analysis, vol. 46, no. 3-4, pp. 275–298, 2006.
15 V. D. R˘adulescu, “Singular phenomena in nonlinear elliptic problems: from blow-up boundary solutions to equations with singular nonlinearities,” in Handbook of Differential Equations: Stationary Partial Differential Equations, M. Chipot, Ed., vol. 4, pp. 485–593, Elsevier/North-Holland, Amsterdam, The Netherlands, 2007.
16 S. Dumont, L. Dupaigne, O. Goubet, and V. R˘adulescu, “Back to the Keller-Osserman condition for boundary blow-up solutions,” Advanced Nonlinear Studies, vol. 7, no. 2, pp. 271–298, 2007.
17 M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, vol. 146 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1991.
18 M. A. Krasnosel’skii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, P. Nordhoff, Groningen, The Netherlands, 1961.
19 C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Boston, Mass, USA, 1988.
20 A. Zygmund, Trigonometric Series. Vols. I, II, Cambridge University Press, New York, NY, USA, 2nd edition, 1959.
