ON THE EXISTENCE OF GLOBAL CLASSICAL ... - Project Euclid

PACIFIC JOURNAL OF MATHEMATICS Vol. 60, No. 2, 1975

ON THE EXISTENCE OF GLOBAL CLASSICAL SOLUTION OF INITIAL-BOUNDARY VALUE 3 PROBLEM FOR D u - u = f YUKIYOSHI EBIHARA, MITSUHIRO NAKAO AND TOKUMORI NANBU

In this paper we shall give a sufficient condition under which an initial-boundary value problem f o r D w - u 3 = / has a global classical solution. 1. I n t r o d u c t i o n . Let Ω C / ? 3 be an open bounded domain with sufficiently smooth boundary dΩ. In this note we are concerned with the existence of a global solution of the initial-boundary value problem: d2 —2 u -Δw + γw3 = /(x, t) at

(*)

for

x E Ω, ί > 0 ,

u(x, 0) = uo(x), -£ u(x, 0) = «,(*), x e Ω, σt and

u(x,t)\dn = 0 for ί ^ O , where Δ is the Laplacian in R3 and γ = - 1. For the equation (*) with γ = + 1 instead of - 1, as is well known, the existence of a global classical solution was proved by J. Sather [4]. His method, however, depends largely on the monotonically increasingness of the term w3, and is not applicable to our problem in its original form. On the other hand D. H. Sattinger [5] introduced the concept of potential well (stable set) to show the existence of global but generalized solutions of the initial-boundary problems of hyperbolic equations with non-monotonic nonlinear terms, though in the case f(x, t) = 0. The method of potential well is useful also for nonlinear partial differential equations of other types (Lions [2], Tsutsumi [6]). Now, a local existence of a classical solution for (*) is known (Ebihara [1]), but that of a global one seems to be unknown and the aim of this note is to give it by combining the method of Sather's with the one of Sattinger's. Roughly speaking our result is: Let {w0, wj belong to the stable set and be sufficiently smooth, and moreover let / have small norm and be smooth. Then (*) admits a global classical solution. 63

64

Y. EBIHARA, M. NAKAO AND T. NANBU

Though we treat only a typical equation with also typical nonlinear term, our method should be applicable to more general ones. 2. P r e l i m i n a r i e s . Throughout this paper the functions considered are all real valued and the notations are as usual (e.g. Lions [2], Mizohata [3]). In this section we offer some lemmas which will be used later. LEMMA 1 (Sobolev).

(i)

// 1 g q g 6, we have

|| u II L 0, the estimations |Df+IM

(li)

for

k = 1,2,3,4

ί G / = [0, Γ],

hold, where CK(T) are constants depending on T but independent of m. Indeed by the linearity of (5) with respect to φk we obtain (12)

(D? + I MO, Ό^um{t)) - (D[ul

+ {{D\um{t\

Dfxum(t))

D\+i um) = (D{/, D{+1 um).

(12) with / = 1 implies

(\um\t\Dtum\l+\D2tum\2)

^ const.

here we used Holder's inequality, Lemma 1 and (10). Applying the GronwalΓs lemma we get (13)

{ | |

| | | | J } e — < κ

τ

for

where (D*« m ) n denotes the value of D , um(x, t) at ί = 0.

ON THE EXISTENCE OF GLOBAL CLASSICAL SOLUTION \\(D,um)o\\ is obviously uniformly b o u n d e d in m. \\(D2,um)0\\, set ί = 0 i n (5) to get 3

((D]um)0-(Aum)o-(u m)0-(f)o,φκ)

69

For the bound of

= 0,

l^k^m,

and hence

where Pm is the orthogonal projection onto the m-dimensional subspace of U with basis {ψu ψ2, - , ψm}. This implies, with the aid of Lemma 1, the uniformly boundedness of \(D] um)0\. Combining these uniform estimates of initial values with (13), we obtain (11) for k = 1. The succession of similar procedure gives (11) for k =2,3,4. Now by the standard arguments of the approximate solutions we conclude, after the extraction of suitable subsequence if necessary, the following: Dktum^Dktu k

in

L2(ίlxl)

k

in

D tum(t)-*D u(t)

for

2

L (Ω)

weakly in

D^Umiή^D^)

0gfcg4,

uniformly for

ίE/,0gfcg3,

2

L , 0=gy, fc^3, t G /,

where D. denotes —- , σXj k

D ,u(t)EHu

0^k^3,

Dk,um(t)->D*u(t)

weakly in

\D*,u(t)-D*,u(τ)\

L\ t E I,

^const.|f-τ|,

k

\D,D ,u(t)-DlDΪ(r)\ S const. | / - τ | , |Dίu B (x, 01 = const. 0 ^ ί c ^ 3 ,

0^/,

fc^3,

(jc,f)GftxJ,

and Dk,ui{t)^Dku\t)

in

L2 uniformly in

t E I,

0gfc^2.

The limit-function u satisfies of course:

for

VvEH\

Vie/,

0siVy^2.

70

Y. EBIHARA, M. NAKAO AND T. NANBU

Moreover applying the well-known regularity results concerning weak solution of elliptic equation, we obtain finally 4

k

k

1

u e C\I, Ho) Π C* (I, H Π H ). k=\

k

Ak

2

Since Π Lo C (I, H (Ω)) C C (Ω x /) holds (c.f. Lemma 1, (ii)), we 2 conclude that u belongs to the class C (fl x /) and is the classical solution on Ωx / of the problem (*). From the arbitrariness of T and the uniqueness of the classical solution on [0, T] (it is obvious) we can construct, as is usual, the classical solution M o n Ω x [0, oo). The proof of theorem is now completed. REFERENCES 1. Y. Ebihara, On the local classical solutions of the mixed problem for nonlinear wave equations. (To appear). 2. J. L. Lions, Quelques mέthodes de resolution de problemes aux limites non lineaires, Dunod, Paris, (1969). 3. S. Mizohata, The theory of partial differential equations, Iwanami, Japan, 1965. 4. J. Sather, The existence of a global classical solution of the initial boundary value problem for D « + u 3 = /, Arch. Rat. Mech. Anal., 22 (1966), 292-307. 5. D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rat. Mech. Anal., 30 (1968), 148-172. 6. M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. R.I.M.S., Kyoto Univ., 8 (1972), 211-229. Received July 17, 1974. FUKUOKA UNIVERSITY, KYUSHU UNIVERSITY, AND KYUSHU UNIVERSITY.