DOI 10.1007/s10958-016-2897-8
Journal of Mathematical Sciences, Vol. 216, No. 2, July, 2016
NECESSARY CONDITIONS FOR STABILIZATION OF SOLUTIONS TO THE DIRICHLET PROBLEM FOR DIVERGENCE PARABOLIC EQUATIONS V. N. Denisov
∗
Moscow State University, Moscow 119991, Russia
[email protected]
A. A. Martynova Vladimir State University 87, ul. Gor’kogo, Vladimir 600000, Russia
[email protected]
UDC 517.9
We establish necessary conditions for the stabilization to zero of solutions to the Dirichlet problem for a parabolic equation with elliptic operator of divergence form and coefficients depending on x and t. Bibliography: 8 titles. In a cylinder Q = D × (0, ∞), we consider the boundary value problem Lu ≡
N ∂ ∂u ∂u = 0 in Q, aik (x, t) − ∂xi ∂xk ∂t
i,k=1
ut=0 = u0 (x),
uS = 0,
(1)
S = ∂D × (0, ∞),
where the initial function u0 (x) is bounded, |u0 (x)| M , x = (x1 , . . . , xN ) ∈ D ⊂ RN , N 2, the coefficients aik (x, t) are bounded and measurable in Q = D × (0, ∞), and D is an arbitrary (possibly unbounded) domain in RN . We assume that aik = aki , i, k = 1, . . . , N , and the uniform parabolicity condition holds: λ20 |ξ|2 (a(x, t)ξ, ξ) λ21 |ξ|2 , λ0 > 0,
λ1 > 0 ∀(x, t) ∈ Q, ξ ∈ D.
(2)
A solution to the boundary value problem (1) is understood in the sense of an integral identity in the function class V˙ 21,0 (QT BR ) for all T > 0, where QT = G × (0, T ), BR is the ball |x| < R in D (cf. [1, 2]). The existence and uniqueness of a bounded solution to the problem (1) is proved in [1]. ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 84, April 2016, pp. 83-88. c 2016 Springer Science+Business Media New York 1072-3374/16/2162-0236
236
We are interested in the pointwise (uniform with respect to x in each compact set K ⊂ RN ) stabilization to zero of the solution to the problem (1), i.e., whether the following limit exists: lim u(x, t) = 0 ∀x ∈ D
t→∞
(∀K ⊂ D).
(3)
In the case where the coefficients aik depend only on x, a criterion for the stabilization to zero of the solution to the problem (1) is established in [2]. In the case where the coefficients aik depend on x and t, sufficient conditions for the stabilization to zero of solutions to the problem (1) are obtained in [3]. In this paper, we obtain (cf. Theorem 1) necessary conditions for the stabilization to zero of the solution to the problem (1). In the case N 3, these necessary conditions coincide with sufficient ones obtained in [3], i.e., we obtain (cf. Theorem 2) a criterion for the stabilization to zero of the solution to the problem (1). In the case N 2, we establish (cf. Theorem 3) a criterion for the stabilization of the solution to the problem (1) in terms of the divergence of integrals of heat capacities. We introduce some definitions. We extend the coefficients of the operator L to the space N +1 , N 2, with preserving the condition (2) by setting aik (x, t) = δi,k , i, k = 1, 2, . . . , N , for R (x, t) ∈ RN +1 \Q. The extended operator is denoted by L. Let Γ(x, y, t, τ ) be the fundamental solution to the extended parabolic operator L. Denote Γ(x, y, t, τ ), t > τ, (4) F (x, y, t, τ ) = 0, t τ. We define the heat capacity of a compact set E ⊂ RN +1 (cf., for example, [4]). A measure μ is called admissible if (5) U μ (x, t) = F (x, y, t, τ )dμ(y, τ ) 1, (x, t) = E. E
The number sup μ, where the supremum is taken over all admissible measures, is called the heat capacity (denoted by γ(E)) of the compact set E ⊂ RN +1 [4]. It is well known (cf., for example, [5]) that the heat capacity of a compact set E ⊂ RN +1 exists. Let E be a compact set in RN , N 3. A measure μ is called admissible if dμ(y) 1, x ∈ RN \E. |x − y|N −2 E
The number sup μ, where the supremum is taken over all admissible measures μ, is called the Wiener capacity (denoted by cap (E) of the compact set E ⊂ RN . Let Qt = Bt × (0, t2 ), where Bt is a ball with center x0 and radius t: Bt = {x : |x − x0 | < t}. Let N 2. The main result of this paper is formulated as follows. Theorem 1. If the solution to the problem (1) stabilizes to zero, then the integral ∞
γ(t)t−N −1 dt = +∞,
t0 > 0,
(6)
t0
is divergent, where γ(t) = γ(Qt \Q) is the heat capacity of the cylinder Qt \Q. 237
We note that c2 t2 cap (Bt \D) γ(Qt \Q) c1 t2 cap (Bt \D),
(7)
where cap (Bt \D) is the Wiener capacity of the compact set Bt \D, c1 > 0, c2 > 0. The left inequality in (7) was proved in [2], and the right inequality was established in [8]. As is proved in [3], if ∞
cap (Bτ \D)τ 1−N dτ = +∞,
t0 > 0,
(8)
t0
for N 3, then the solution to the problem (1) stabilizes to zero, i.e. the limit (3) exists. Taking into account the inequality (7) and Theorem 1, for N 3 we obtain the following criterion for the stabilization to zero of the solution to the problem (1). Theorem 2. The solution to the problem (1) stabilizes to zero if and only if the integral (8) is divergent. The authors do not know whether the estimate (7) is valid for the heat capacity γ(Qt \Q) in the case N = 2. Theorem 3. Let N 2. The solution to the problem (1) stabilizes to zero if and only if the integral (6) is divergent. In Theorems 1–3, a point x0 can be arbitrary since the choice of this point does not affect whether the integrals (6) and (8) are convergent or divergent. To prove the theorems, we need to estimate from below the heat capacity of the cylinder in terms of the Wiener capacity of the base multiplied by the height of the cylinder. For this purpose we use the estimates proved in [3, 5]. To prove Theorem 3, we need to study properties of parabolic capacities and the corresponding heat potentials for Equation (1). As is known, for the fundamental solution Γ(x, y, t, τ ) of the operator L the following twosided Aronson estimate holds [6]: N
c2 (t − τ )− 2 exp
−
N r2 r2 Γ(x, y, t, τ ) c1 (t − τ )− 2 exp − , 4β(t − τ ) 4α(t − τ )
(9)
where r = |x − y| and the constants c1 , c2 , α, and β depend only on N , λ0 , and λ1 . Without loss of generality we assume that the origin is located on the boundary of a domain D ⊂ RN and x0 = 0. Let Br be an open ball of radius r and center at the origin. We consider a sequence of cylinders Qi = B2i × (t0 − 4i , t0 ), i = m0 , m0 + 1, . . . , m1 , m1 + 1, . . ., where t0 = 4m1 , and denote Em0 = (RN +1 \Q) ∩ Qm0 , Hi = (RN +1 \Q) ∩ Qi ,
Ei = (RN +1 \Q) ∩ (Qi \Qi−1 ),
i = m0 + 1, m0 + 2, . . .
i = m0 , m0 + 1, . . .
i = (RN +1 \Q) ∩ ((B 2i \B2i−1 ) × (0, t0 )), E
(11) i = m1 + 1, m1 + 2, . . . .
To prove Theorem 2, it suffices to prove the following assertion. 238
(10)
(12)
Lemma 1. If
∞
N
4− 2 i γ(Hi ) < ∞,
(13)
i=m0
then the solution to the problem (1) does not stabilize to zero as t → ∞, i.e., there exists a sequence of points (xk , tk ), xk ∈ D ∩ Br0 , tk → ∞, such that u(xk , tk ) > const. Indeed, let us consider the series ∞ γ(Hi ) , qN i
q > 1.
(14)
i=m0
We note that (14) becomes (13) if q = 2. The following assertion is proved in [7, Lemma 2.2]. Lemma 2. The series (14) is convergent (divergent) if and only if the integral ∞ t0
γ(t) dt, tN +1
t0 > 0
(15)
is convergent (divergent), where γ(t) = γ(Qt \Q), Qt = Bt × (0, t2 ). Therefore, the convergence (divergence) of the integral (15) and the series (14) are equivalent. Proof of Lemma 1. Let the measure μi realizes the capacity of the compact set Ei , i =
i so that U μi (x, t) = 1 for the potential 1, 2, . . . , m1 , and the capacity of the compact set E μi U (x, t) = F (x, y, t, τ )dμ(y, τ ) Ei
i . Let of this measure, where (x, t) ∈ Ei or (x, t) ∈ E m2
U (x, t) =
U μi (x, t),
(16)
i=m0
where the constant m2 > m1 will be chosen below. We note that U (x, t) 1,
(x, t) ∈ (∂D ∩ B2m2 ) × (0, t0 ),
U (x, t) = 0,
t = 0.
We introduce the auxiliary function
(17)
V (x, t) = K ·
F (x, y, t + t0 , 0)dSy
(18)
|y|=2m2
in the cylinder B2m2 × (0, t0 ). We apply the lower Aronson estimate (9) to (18) and choose a large constant K > 0 such that V 1 on the lateral surface of the cylinder. For this purpose we note that |x − y|2 Kc2 |x − y|2 Kc2 V (x, t) dSy exp − exp − dSy N N 4β(t + t0 ) 4βt0 (t + t0 ) 2 (2t0 ) 2 |y|=2m2
|y|=2m2
239
for 0 < t < t0 . Further, for |x| = 2m2 we have |x − y|2 exp − > a, 4βt0 |y|=2m2
where a is a positive number independent of R = 2m2 (cf. [4]). Therefore, setting N
(2t0 ) 2 K= , ac2 we obtain V 1 on the lateral surface of the cylinder. Summarizing the above facts about the function V (x, t), we conclude that V (x, t) 1,
(x, t) ∈ (D ∩ ∂B2m2 ) × (0, t0 ),
V (x, t) 0,
t = 0.
(19)
By the maximum principle, from (17) and (19) it follows that 1 − u(x, t) U (x, t) + V (x, t),
(x, t) ∈ Qm2 ,
(20)
where u(x, t) is the solution to the problem (1) and the functions U and V are defined in (16) and (18) respectively. Let x be an arbitrary point of the domain D ∩ B2m0 , and let m2 be large enough so that |x | <
R = 2m2 −1 . 2
Using the upper Aronson estimate (9), we find N 1 R2 V (x , t) < Kc1 exp − dS (t0 )− 2 c1 , N 8αt0 (2t0 ) 2 |y|=R
C(N )KRN −1 exp
−
R2
32αt0
ε,
m2 l
(21)
∀x ∈ D ∩ B2m0 ,
where the constant l depends only on N , t0 , λ0 , and λ1 . Let us estimate U μi (x, t) from above for i > m0 at an arbitrary point (x , t) of the set (D ∩ B2m0 −1 ) × t0 . It is easy to see that N
U μi (x , t0 ) C4− 2 i γ(Ei ), N
i ), U μi (x , t0 ) C4− 2 i γ(E
i = m0 + 1, m0 + 2, . . . , m1 i = m1 + 1, m1 + 2, . . . , m2 ,
where the constant C depends only on c1 , λ0 , λ1 , and N . Since E i ⊂ Hi ,
i ⊂ Hi , E
(22)
from (10)–(12) we find m2 i=m0 +1
240
μi
U (x , t0 ) C
m2 i=m0 +1
N
4− 2 i γ(Hi ).
(23)
Since the series (13) is convergent, we can choose a large constant m0 such that ∞
N
4− 2 i γ(Hi ) < ε,
(24)
i=m0
where the constant ε > 0 will be chosen below. Hence N
4− 2 m0 γ(Hm0 ) < ε.
(25)
U (x , t0 ) Cε ∀x ∈ D ∩ B2m0 −1 .
(26)
By (23) and (24), we have
We show that ε ε0 , where the constant ε0 depends only on N , λ0 , λ1 and there exists a point x ∈ D ∩ B2m0 −1 such that 1 (27) U μm0 (x , t0 ) < . 8 Assume the contrary. Suppose that T = (D∩B2m0 −1 )×(t0 , t0 +4m0 ), v(x, t) is the solution to the problem, Lv = 0 in T , v ∂(D∩B m −1 )×(t0 ,t0 +4m0 ) = U μm0 , v t=t0 = 1/8. Since U μm0 (x, t) 1/8 2
0
for x ∈ D ∩ B2m0 −1 by assumption, from the maximum principle it follows that v(x, t) U μm0 (x, t)
in T.
(28)
We extend the solution v(x, t) to the cylinder (D ∩ B2m0 −1 ) × (t0 − 4m0 , t0 + 4m0 ) by setting v(x, t) = 1/8 for (x, t) ∈ (D ∩ B2m0 −1 ) × (t0 − 4m0 , t0 ). By the Harnack inequality, we have v(x, t0 + 4m0 ) c0 ,
x ∈ D ∩ B2m0 −2 ,
where the constant c0 depends only on N , λ0 , and λ1 . By (28), we have U μm0 (x, t0 + 4m0 ) c0 ,
x ∈ D ∩ B2m0 −2 .
(29)
On the other hand, a simple computation with (25) taken into account shows that N
U μm0 (x, t0 + 4m0 ) c3 4− 2 m0 γ(Hm0 ) c3 ε,
x ∈ D ∩ B2m0 −2 ,
where the constant c3 depends only on N , λ0 , and λ1 , which contradicts (29) for sufficiently small ε. Setting x = x , t = t0 in (20), where x is the point in (27), and taking into account (20) and (21), for sufficiently small ε > 0 and a suitable choice of m0 , we get 1 1 − u(x , t0 ) U (x , t0 ) + V (x , t0 ) . 4 Consequently, u(x , t0 ) 3/4. Since t0 can be so large as desired, the solution to the problem (1) cannot stabilize to zero as t → ∞ in a fixed cylinder D ∩ B2m0 −1 × (0, ∞). The lemma is proved. The proof of Theorem 1 is obtained from Lemma 1 since, under the assumptions of the theorem, the integral (15) is divergent which, in view of the estimate (7), is equivalent to the divergence of the integral (6) or the divergence of the series (14). Theorem 2 is valid since we can establish the necessity from Theorem 1 and the estimate (7) whereas the sufficiency follows from [3, Theorem 2]. Theorem 3 is valid for N 2 since the necessity follows from the proof of Theorem 1 in view of Lemma 2, whereas the sufficiency is established in the same way as in Theorem 1.2 of [7]. 241
Acknowledgments The work is financially supported by the Russian Foundation for Basic Research (project No. 15-01-00471). The authors thank Professor Yu. A. Alkhutov for useful discussions.
References 1.
F. Kh. Mukminov, “On uniform stabilization of solutions of the first mixed problem for a parabolic equation” [in Russian], Mat. Sb. 181, No. 11, 1486-1509 (1990); English transl.: Math. USSR, Sb. 71, No. 2, 331–354 (1992).
2.
V. N. Denisov, “Necessary and sufficient conditions of stabilization of solutions of the first boundary value problem for a parabolic equation” [in Russian], Tr. Semin. Im I. G. Petrovskogo 29, No. 2, 248–280 (2013); English transl.: J. Math. Sci., New York, 197, No. 3, 303–324 (2014).
3.
V. N. Denisov, “Conditions for stabilization of solution to the first boundary value problem for parabolic equations” [in Russian], Probl. Mat. Anal. 58, 143–159 (2011); English transl.: J. Math. Sci., New York, 176, No. 6, 870-890 (2011).
4.
E. M. Landis, Second Order Equations of Elliptic and Parabolic Type [in Russian], Nauka, Moscow (1971); English transl.: Am. Math. Soc., Providence, RI (1998).
5.
Yu. A. Alkhutov, “Removable singularities of solutions of second order parabolic equations” [in Russian], Mat. Zametki 50, No. 5, 9–17 (1991); English transl.: Math. Notes 50, No. 5, 1097–1103 (1991).
6.
D. G. Aronson, “Non-negative solutions of linear parabolic equations,” Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser 22, No. 4, 607–694 (1968).
7.
V. N. Denisov and A. A. Martynova, “Necessary and sufficient conditions for the stabilization of a solution to the Dirichlet problem for parabolic equation” [In Russian], Probl. Mat. Anal. 68, 81–88 (2013); English transl.: J. Math. Sci., New York 189, No. 3, 422–430 (2013).
8.
W. P. Ziemer, “Behavior at the Boundary of Solutions of Quasilinear Parabolic equations,” J. Differ. Equ. 35, 291-305 (1980).
Submitted on February 18, 2016
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