Pacific Journal ofMathematics

ASYMPTOTIC STABILITY AND NONEXISTENCE OF GLOBALSOLUTION FOR A SEMILINEAR PARABOLIC EQUATION

CHIA-VEN PAO

Vol. 84, No. 1 May 1979

PACIFIC JOURNAL OF MATHEMATICSVol. 84, No. 1, 1979

ASYMPTOTIC STABILITY AND NON-EXISTENCE OFGLOBAL SOLUTION FOR A SEMI-LINEAR

PARABOLIC EQUATION

C. V. PAO

The aim of this paper is to investigate the existence andnonexistence of a nonnegative global solution for a semilinearparabolic system in a bounded domain. It is shown thatfor a certain class of initial functions the correspondingsolution of the initial boundary value problem has a finiteescape time, while for another class of initial functions aunique solution exists for all time and diminishes to zero.This result leads to an explicit estimate for the stabilityand the instability regions of the trivial steady-state solution.In the case of nonexistence of global solutions, an estimatefor the finite escape time is also given.

I Introduction. The problem of nonexistence of global solutionfor the semilinear parablic equation

(1.1) ut-V-(D(x)Vu) = au1+a(t e (0, T], xeΩ)

has been given considerable attention in recent years (cf. [4-6, 13,15]). In most of the above references the spatial domain Ω wastaken as the whole space R* so that the system under considerationby these authors is a Cauchy problem. In this paper, we considera bounded domain Ω in Rn together with the following boundaryand initial conditions:

(1.2) β— + u = 0 (t 6 (0, T], x 6 3Ω)

(1.3) u(0, x) = nix) {x 6 Ω) .

In the above system, V is the gradient operator, D is a positivefunction on Ω Ξ Ω U dΩ, a, a, β are constants with a > 0, β 2> 0, dΩis the boundary of Ω and v is the outward normal vector on diλThis system arises from models of simultaneous diffusion and recom-bination of electrons and irons as well as models of reactor dynamicswhere positive feedback is allowed when a > 0 (cf. [1, 8, 15]). Weassume that Ω is sufficiently smooth, D is continuously differentiatein Ω and u0 is continuous nonnegative on Ω and satisfies the boundarycondition (1.2). The constant, α, may assume positive or negativevalues and its magnitude plays an important role in relation to thestability region of the trivial steady state solution.

191

192 C. V. PAO

The nonexistence of global solution to the Cauchy problem (1.1)and (1.3) for the case a = D = 1 in the whole space Rn has beendiscussed by Fujita [4, 5], Hayakawa [6], Sugitani [16] and morerecently by Portnoy [13]. A similar problem for a bounded domainΩ has been investigated by Kaplan [7], Friedman [2] and Pao [11].(See also Levine and Payne [9].) The purpose of this paper is toinvestigate the existence and nonexistence of a nonnegative globalsolution to the system (1.1)-(1.3) and to give a threshold result onthe asymptotic behavior of the solution. Specifically, we show thatfor a certain class of initial functions u0 a unique nonnegative solutionu(t, x) to (1.1)-(1.3) exists on some finite interval [0, Γo) and

lim(max u(t, x)) = ^ as t > To

and for another class of initial functions, the solution u exists on[0, °o) and \im u(t, x) = 0 on Ω as t —• ©o. Moreover, we establish inthe first case an explicit upper bound for the finite escape time To

while in the second case we give an estimate for the stability regionas well as the instability region of the zero solution. We also showthat when a <; 0 the nonnegative solution u diminishes to zero forevery nonnegative initial function u0. Thus if we increase the valueof a from negative to positive then the zero steady state solutionchanges form global asymptotic stability to regional asymptoticstability; and for initial functions outside certain region the corre-sponding solutions diverge to infinite in finite time. As is to beexpected our estimate demonstrates that larger initial function u0

requires less time for the blowing-up property of the solution. Hencethe consideration of a bounded domain instead of the whole spaceRn leads to more delicate asymptotic behavior of the solution. Itis interesting to note that this qualitative behavior also has a simpleand natural physical interpretation and the results can be extendedto more general systems (see the remarks in §2). In the presentpaper, however, we limit our discussion to the system (1.1)-(1.3).

2* The main results* In order to obtain our main results onthe existence and nonexistence of a global solution we as in [10, 11]use the notion of upper and lower solutions for parabolic systems.By an upper solution we mean a smooth function u(t, x) satisfyingthe inequalities

ut~F- {D{x)Vn) au1+a (t e (0, T], xeΩ)

(2.1) β— + u^0 (te (0, Γ], x e dΩ) ,

S(0, x) ^ uo(x) (x 6 Ω)

ASYMPTOTIC STABILITY AND NON-EXISTENCE OF GLOBAL SOLUTION 193

and a lower solution is a smooth function u(t, x) which satisfies allthe reversed inequalities in (2.1). Moreover we require that u <^ uon [0, T] x Ω. Here by a smooth function it meant a continuousfunction w on [0, T] x Ω which is continuously differentiate in ί e(0, T], twice continuously differentiate in x e Ω and du/dv exists on(0, T] x dΩ.

Suppose that upper and lower solutions u, u exist and u <* u.Then by starting from the initial iteration uw = u and u{0) = u,respectively, we can construct two sequences from the recurencerelation

uik) - F(D(x)Fu{k)) =-- a{u[k~ι))lΛtx (t e (0, T], x e Ω)

(2.2) dv

u{k)(0, x) = uo(x) (x e Ω)

k = 1, 2, .

Denote these two sequences, respectively, by {ΰ{k)} and {u{k)} (so thatύ{0) = ίϊ, ^ ( 0 ) = ύ). Then by the properties of upper and lower solu-tions one can easily show that (i) ΰ{1c+1) <: ΰ{k\ (ii) u{k) ^ ySk+1) and(iii) u{k) ^ iZ(Z:) for every k = 0, 1, 2, . These inequalities imply thatthe pointwise limits

Km u{k) = ΰ and lim u{k) = u as & > co

exist and satisfy the relation u <. y, ύ ^ ϊί on [0, T] x Ω. In fact,y, coincides with ΰ and is the unique solution of (1.1)-(1.3) (cf.[10, 12]). Therefore if we can find a pair of upper and lower solutionwith u <* u, then a unique solution u(t, x) to (1.1)-(1.3) exists and

(2.3) u(t, x) ^ u(t, x) g fi(ί, x) on [0, Γ] x Ω .

Since for u0 ^ 0 the function u — 0 satisfies all the reversed inequa-lities in (2.1), the existence of a nonnegative solution u satisfying0 <J u(t, x) ^ u(t, x) is insured if one can find an upper solution ί£which is nonnegative. We shall do this for our global existenceproblem.

In the proof of the theorems the construction of u is based onthe linear eigenvalue problem

V (D(x)Fφ) + Xψ = 0 (xe Ω)

( 2 4 ) β*t + φ = o (xedΩ).dv

It is well-known that the least eigenvalue λ0 is positive and its cor-responding eigenfunction ψ(x) is positive in Ω. In fact, if β > 0 the

194 C. V. PAO

maximum principle implies that ψ(x) > 0 on Ω. We normalize ψ sothat max ψ(x) = 1. For convenience we set ψm = min^r(x). Noticethat ψm > 0 when β > 0. Our main results are stated in the follow-ing two theorems.

THEOREM 1. Let β > 0, a > 0 and η = (λo/α)17^-1. Then for anyuo(x) ^ bf(x) with b > η there exists a constant JΓ0 < °° such that aunique solution u(t, x) to (1.1)-(1.3) exists on [0, To) x Ω and satisfies

(2.5) lim max u(t, x) = oo .

Moreover,

(2.6) Γo ^ (ax,)-1 In [α(6f J7(α(6f J α - λ0)] .

THEOREM 2. Let Xe (0, λ0) αwd ^ = ((λ0 - X)/a)ι/a when a > 0./or ατzi/ uo(ίc) <; pλψ(x), a unique nonnegative solution u(t, x) to

(1.1)-(1.3) exists on [0, <^) x Ω and satisfies

(2.7) 0 ^ u(ί, a;) Pxe~uf(x) (t> 0,xeΩ) .

In the case of α ^ 0 the above inequality holds for every px =p < co whenever wo(x) ^ pψ(x).

REMARK, (a) The results in Theorems 1 and 2 show that whena > 0 a stability region of the zero solution is given by the set Λt =={wo; 0 <; w0 ^ ^ψ4} while an instability region is Λ2 = {uo; uQ ^ 6^}. Bychoosing λ sufficiently small the value of pλ may be taken arbitrarilyclose to (λo/α)1/α. Since λ0 is directly related to the size (for a fixedgeometry) of the domain Ω it follows that smaller domain tendsto stabilize the equilibrium solution u — 0 while large domain tendsto destabilize the zero solution. This is, of course, to be expectedphysically since energy leaks faster in smaller domains than largerones.

(b) Theorems 1 and 2 remain true when the operator V (D(x)Fu)is replaced by a more general uniformly elliptic operator in the form

n n

Lu = ^ aiό(x)ux.xj + Σ blx)ux. .

In this situation the least eigenvalue λ0 and its corresponding eigen-f unction ψ should be with respect to the operator L (under the sameboundary condition). Notice that λ0 > 0 (or at least there exists aeigenvalue λ0 which is real positive) and the corresponding eigen-function ψ is positive (cf. [14]).

The results in Theorems 1 and 2 can be given a physical inter-

ASYMPTOTIC STABILITY AND NON-EXISTENCE OF GLOBAL SOLUTION 195

pretation such as in the process of diffusion or heat conduction.When the spatial domain is bounded, energy (or heat) leaks throughthe boundary surface dΩ. Thus for the class of initial functionso ^ pxψt energy leaks away through the boundary surface before

it gets large so that the effect due to the feedback term au1+a (whichacts as a source for a > 0) is diminishing. But when the initialfunction u0 is large to the extent of u0 > ηψ, the feedback termdominates the leakage initially and its effect tends to grow andeventually leads to the unboundedness of the solution in finite time.It is interseting to note that our estimate indicates that the finite escapetime To is larger for smaller values of a and becomes smaller as agets larger. In fact, if we increases the value of a from α ^ O toa > 0 then the system (1.1)-(1.3) changes from global asymptoticstability to regional asymptotic stability, and as a increases, thestability region gets smaller while the instability region becomeslarger.

It is to be noted that there remains a gap between the stabilityregion Λx and the instability region Λ2 (by a factor of ψς;1)- This gapcan be closed if one can show that for uQ not in these two regionsthe corresponding solution lies either in Λλ or in Λ2 after a finitetime.

3» Proofs of the main theorems*

Proof of Theorem 1. We first construct a function u satisfyingthe reversed inequalities in (2.1). Let u = e~^p(t)f (x) with p(0) < b,where p(t) is a continuously differentiable function to be chosen.Since u(0,x) = p(fi)ψ(x) <>uo(x) and by (2.4) A (Dμu) = ~\ou andβdujdv + u — 0 we see that u is a lower solution if

(3.1)

where Pf = dP/dt. Clearly, the above inequality holds if

(3.2) P ' ^ a^le-aλQtPί+a .

It is easily seen that the function P(ί) given by

(3.3) P(t) = [b~a + αλo-tyi(l - e-«*o*)]-i/« , (p e [0, TO)

satisfies the requirement in (3.2) for t e [0, 2\), where 2\ is given bythe right side of (2.6). Hence the function u = e~λotP(t)ψ(x) with P(ί)given by (3.3) is a lower solution on [0, T2] x Ω for every T2 < Tλ.Using this function as the initial iteration in (2.2) with T < Tx weobtain a monotone nondecreasing sequence which converges to asolution u of (1.1)-(1.3) on [0, T] x Ω provided that this sequence is

196 C. V. PAO

uniformly bounded from above. However, this may not be the casefor every T < Tx unless u has the finite escape time exactly at 2\.This is due to the fact that u is a lower bound of u when the solutionexists. To overcome this difficulty we take an arbitrarily largeconstant M > uo(x) and define a function f(u) by

(3.4) f(u) = au1+a when u <> M and f(u) = aM1+a when u > M .

Choose T2 sufficiently close to 2\ such that u(T2, x) ^ M for somex e Ω. Then by replacing the right-side of the first equation in (2.2)by /(tt1*""") and using the initial iteration u{0) = u with T = T2 weagain obtain a monotone nondecreasing sequence for the "modified"system (1.1)-(1.3) (i.e., with au1+a repalced by f(u)). We denote thissequence also by {u{k)}. Now since f{Vik)) is uniformly bounded, anapplication of the well-known estimate for linear parabolic systemsimplies that {u{k)} is bounded (e.g., see [3] p. 146). It follows fromthe monotone property that {u{k)} converges to a unique solutionu*(t, x) of the modified system (1.1)-(1.3) and

(3.5) v,*(t, x) ^ u(t, x) = e-^P{t)f{x) (t 6 [0, ΓJ x Ω) .

But u(T2, x) ^ M for some x e Ω and uQ(x) < M, there exists T3 ^ T2

such that u*(t, x) g M for (ί, x) e [0, Γ3] x Ω and w*(Γ3, a?) = M forsome x 6 β. By the definition of f(u), u* is the solution of (1.1)-(1.3)with T — Tz. Since M can be chosen arbitrarily large the aboveconclusion implies that the solution of (1.1)-(1.3) must be unboundedon [0, To] x Ω for some To ^ Tlβ For if it were bounded (say, by K)on [0, T] x Ω for every T ^ TΊ, then by choosing M > K in thedefinition of /(w) we obtain a solution ^* of (l.l)-(l-3) such thatu*(t, x) = ikf at some point (ί, a?) e [0, T8] x Ω where T3 < Tx. Clearlythis is impossible. The proof of the theorem is completed.

Proof of Theorem 2. For the case a > 0 it suffices to show thatu Ξ= pλe~~uψ(x) is an upper solution on [0, T] x Ω for every finite T.In view of (2.1) and since the last two conditions are satisfied, thiswill follow if

(3.6) (λ0 - \)p2e-λtψ(x) ^ a(ρλe~λi<φ{x))1+a .

The above inequality is clearly satisfied for any px satisfying pa

λ <£(λ0 — λ)/α. This proves (2.7). When a <; 0, the sequence obtainedfrom (2.2) with uw = u = pe~λt^{x) is no longer monotone nonincre-asing and thus we need to modify our construction. To achieve this,we replace the first equation in (2.2) by

uw -V {D(x)Vu{k)) + Qu{k)

( 8 # 7 ) = a(u{k-ί))ι+a + Qw(fc"1} , (t 6 (0, T], α? 6 β )

ASYMPTOTIC STABILITY AND NON-EXISTENCE OF GLOBAL SOLUTION 197

where Q is a constant satisfying Q ^ ~apa while T < --> is arbitrary.It is easily seen by using the initial iteration u{0) = pe~λtψ(x) that thesequence constructed from (3.7) together with the last two equationsin (2.2) is nonnegative and is monotone nonincreasing (cf. [10]).Therefore it converges to a unique solution u(t, x) of the system(1.1)-(1.3) such that 0 ^ (£, x) g pertλγ(x) on [0, T]xΩ. This provesthe theorem.

REFERENCES

1. C. Y. Chan, Existence, uniqueness, upper and lower bounds of solutions of nonlinearspace-time nuclear reactor kinetics, SIAM J. Appl. Math.. 27 (1974), 72-82.2. A. Friedman, Remarks on nonlinear parabolic equations, Proc. Amer, Math. Soc;Symp. Appl. Math., 17 (1965), 3-23.3. f Partial Differential Equations of Parabolic Type, Prentice Hall, EnglewoodCliffs, N.J., 1964.4. H. Fujita, On the blowing up of solutions of the Cauchy problem for ut — Δu +u1+a, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.5. , On some nonexistence and nonuniqueness theorems for nonlinear parabolicequations, Proc. Symp. Pure Math. Vol. XVIII, Part 1, Amer. Math. Soc, (1970), 105-113.6. K. Hayakawa, On nonexistence of global solutions of some semi-linear parabolicdifferential equations, Proc. Japan Acad., 49 (1973), 503-505.7. S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm.Pure Appl. Math., 16 (1963), 305-330.8. W. E. Kastenberg and P. L. Chambre, On the stability of nonlinear space-dependentreactor kinetics, Nucl. Sci. Engrg., 31 (1968), 67-79.9. H. A. Levine and L. E. Payne, Nonexistence of global weak solutions for classes ofnonlinear wave and parabolic equations, J. Math. Anal. App., 55 (1976), 329-334.10. C. V. Pao, Positive solutions of a nonlinear boundary value problem of parabolictape, J. Differential Equations, 22 (1967), 145-163.11. f Nonexistence of global solutions and bifurcation analysis for a boundary-value problem of parabolic type, Proc. Amer. Math. Soc, (to appear).12. , Successive approximations of some nonlinear initial boundary valueproblems, SIAM J. Math. Anal., 46 (1974), 91-102.13. S. L. Portnoy, On solutions to ut ~ Δu-[-u2 in two dimensions, J. Math. Anal.Appl., 55 (1976), 291-294.14. M. H. Protter and H. F. Weinberger, On the spectrum of general second orderoperators, Bull. Amer. Math. Soc, 72 (1966), 251-255.15. G. Rosen, Solutions to the nonlinear recombination equation for the infinite spatialdomain, SIAM J. Appl. Math., 29 (1975), 146-151.16. S. Sugitani, On non-existence of global solutions for some nonlinear integral equa-tions, Osaka J. Math., 12 (1975), 45-51.

Received March 4, 1977.

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UNIVERSITAT GRAZ

GRAZ, AUSTRIA

PACIFIC JOURNAL OF MATHEMATICS

EDITORSDONALD BABBITT (Managing Editor)

University of CaliforniaLos Angeles, California 90024

HUGO ROSSI

University of UtahSalt Lake City, UT 84112

C. C. MOORE and ANDREW OGG

University of CaliforniaBerkeley, CA 94720

J. DUGUNDJI

Department of MathematicsUniversity of Southern CaliforniaLos Angeles, California 90007

R. F I N N AND J. MILGRAM

Stanford UniversityStanford, California 94305

E. F. BECKENBACH

ASSOCIATE EDITORSB. H. NEUMANN F. WOLF K. YOSHIDA

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Pacific Journal of MathematicsVol. 84, No. 1 May, 1979

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of representations of compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Douglas Michael Campbell and Vikramaditya Singh, Valence properties of

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