Oscillations of solutions of superlinear-sublinear parabolic equations are studied, and the unboundedness of solutions is also investigated as corollaries. The approach used here is to use the Piconetype inequalities for elliptic operators

Jaroš Jaroslav

Kusano Takaŝi

Yoshida Norio

application/pdfOscillations of solutions of superlinear-sublinear parabolic equations are studied, and the unboundedness of solutions is also investigated as corollaries. The approach used here is to use the Piconetype inequalities for elliptic operators.ArticleMathematics journal of Toyama University, Vol.24, Page 83-91departmental bulletin pape

Jaroš, Jaroslav

Kusano, Takaŝi

Yoshida, Norio

University of Toyama Repository

Math. J. Toyama Univ.Vol. 24(2001), 83-91Oscillatory properties of solutions ofsuperlinear-sublinear parabolic equationsvia Picone-type inequalitiesJaroslav Jarosˇ, Kusano Takasˆi and Norio YoshidaAbstract. Oscillations of solutions of superlinear-sublinear parabolicequations are studied, and the unboundedness of solutions is also in-vestigated as corollaries. The approach used here is to use the Picone-type inequalities for elliptic operators.In 1962, McNabb [8] established criteria for unboundedness of solutionsof linear parabolic equations on the basis of Picone-type identities. Hisresults were extended by Dunninger [4], Kusano and Narita [7] to parabolicdifferential inequalities, and by Chan [1], Chan and Young [2, 3], Kuks [6]to time-dependent matrix differential inequalities. All of them also containthe results about zeros of solutions or singularities of matrix solutions.Recently Jarosˇ, Kusano and Yoshida [5] established Picone-type inequali-ties which connect a linear elliptic operator with an associated superlinear-sublinear elliptic operator. By extending the Picone-type inequalities toparabolic equations with time-dependent coefficients, we obtain the oscil-latory behavior or the unboundedness of solutions of superlinear-sublinearparabolic equations.We are concerned with the oscillatory behavior of solutions of the non-2000 Mathematics Subject Classification. 35B05.Key words and phrases. Oscillatory properties, parabolic equations, Picone-typeinequalities.84 Jaroslav Jarosˇ, Kusano Takasˆi and Norio Yoshidalinear parabolic equation∂v∂t− L[v] = 0, (x, t) ∈ Ω ≡ G× (0,∞), (1)where G is a bounded domain in Rn with piecewise smooth boundary ∂GandL[v] ≡n∑i,j=1∂∂xi(Aij(x, t)∂v∂xj)+ C(x, t)|v|β−1v +D(x, t)|v|γ−1v.It is assumed that :(A1) Aij(x, t) ∈ C(Ω;R) (i, j = 1, 2, ..., n) and the matrix (Aij(x, t)) issymmetric and positive definite in Ω ;(A2) C(x, t) ∈ C(Ω; [0,∞)) and D(x, t) ∈ C(Ω; [0,∞)) ;(A3) β and γ are constants such that β > 1 and 0 < γ < 1.Definition 1. The domain DL(Ω) of L is defined to be the set of all func-tions v of class C1(Ω;R) with the property that Aij(x, t)∂v∂xj∈ C1(Ω;R)∩C(Ω;R).Definition 2. By a solution of (1) we mean a function v ∈ DL(Ω) whichsatisfies the equation (1).Definition 3. A solution v of (1) is said to be oscillatory on Ω if v has azero on G× [t,∞) for any t > 0.We consider the linear differential operator ` defined by`[u] =n∑i,j=1∂∂xi(aij(x)∂u∂xj)+ c(x)u, (2)where the coefficients aij(x) and c(x) satisfy the following assumptions :(A4) aij(x) ∈ C(G;R) (i, j = 1, 2, ..., n) and the matrix (aij(x)) is symmet-ric and positive definite in G ;Oscillatory properties of solutions of parabolic equations 85(A5) c(x) ∈ C(G;R).Definition 4. The domain D`(G) of ` is defined to be the set of all func-tions u of class C1(G;R) with the property that aij(x)∂u∂xj∈ C1(G;R) ∩C(G;R).The following theorem is due to Jarosˇ, Kusano and Yoshida [5, Theo-rem 8].Theorem 1. (Picone-type inequality) Assume that u ∈ D`(G), v ∈DL(Ω) and v 6= 0 in G × I, where I is any interval in R. Then we ob-tain the following inequalityn∑i,j=1∂∂xi(uaij(x)∂u∂xj− u2vAij(x, t)∂v∂xj)≥n∑i,j=1(aij(x)−Aij(x, t)) ∂u∂xi∂u∂xj+(β − γ1− γ(β − 11− γ) 1−ββ−γC(x, t)1−γβ−γD(x, t)β−1β−γ − c(x))u2+n∑i,j=1Aij(x, t)(v∂∂xi(uv))(v∂∂xj(uv))+uv(v`[u]− uL[v]) (3)for (x, t) ∈ G× I.Theorem 2. Assume that (A1) – (A5) hold, and that there exists a non-trivial function u ∈ D`(G) such that`[u] = 0 in G,u = 0 on ∂G,limt→∞∫ tTV [u](s) ds =∞ for any T > 0,whereV [u](t) ≡∫G[n∑i,j=1(aij(x)−Aij(x, t)) ∂u∂xi∂u∂xj+(β − γ1− γ(β − 11− γ) 1−ββ−γC(x, t)1−γβ−γD(x, t)β−1β−γ − c(x))u2]dx.86 Jaroslav Jarosˇ, Kusano Takasˆi and Norio YoshidaLet v ∈ DL(Ω) be a solution of (1) which is nonoscillatory on Ω. Then vsatisfieslimt→∞∫Gu2 log |v| dx =∞. (4)Proof. Since v ∈ DL(Ω) is a nonoscillatory solution on Ω, we see thatv 6= 0 on G× [t0,∞)for some t0 > 0. Without loss of generality we may assume that v > 0 onG× [t0,∞). Integrating the Picone-type inequality (3) over G, we obtain0 ≥ V [u](t) +∫Gn∑i,j=1Aij(x, t)(v∂∂xi(uv))(v∂∂xj(uv))dx−∫Gu2vL[v] dx≥ V [u](t)−∫Gu2v∂v∂tdx, t ≥ t0. (5)We see from (5) that∫Gu2∂∂tlog v dx ≥ V [u](t), t ≥ t0and thereforeddt∫Gu2 log v dx ≥ V [u](t), t ≥ t0. (6)Integration of (6) over [t0, T ] yieldsz(T )− z(t0) ≥∫ Tt0V [u](s) ds, (7)wherez(t) ≡∫Gu2 log v dx.Letting T → ∞ in (7), we observe that limT→∞z(T ) = ∞. This completesthe proof.Oscillatory properties of solutions of parabolic equations 87Corollary 1. Assume that (A1) – (A5) hold, and that there exists a non-trivial function u ∈ D`(G) such that`[u] = 0 in G,u = 0 on ∂G,limt→∞∫ tTV [u](s) ds =∞ for any T > 0.Then every bounded solution v ∈ DL(Ω) of (1) is oscillatory on Ω.Proof. Let v ∈ DL(Ω) be any bounded solution of (1). Then, we observethat log |v| is bounded from above, and hence ∫G u2 log |v| dx is also boundedfrom above. Therefore (4) does not hold. Theorem 2 implies that thebounded solution v is oscillatory on Ω.Corollary 2. Assume that the same hypotheses as those of Theorem 2hold. If v ∈ DL(Ω) is a solution of (1) which is nonoscillatory on Ω,then v is unbounded in Ω.Proof. Since v is nonoscillatory on Ω, v satisfies the condition (4). Hence,|v| cannot be bounded from above in Ω, that is, v is unbounded in Ω.The following theorem was established by Jarosˇ, Kusano and Yoshida [5,Theorem 7].Theorem 3. Let v ∈ DL(Ω) and let v 6= 0 in G×I, where I is any intervalin R. Then the following inequality holds for any u ∈ C1(G;R) :n∑i,j=1Aij(x, t)(v∂∂xi(uv))(v∂∂xj(uv))+n∑i,j=1∂∂xi(u2vAij(x, t)∂v∂xj)≤n∑i,j=1Aij(x, t)∂u∂xi∂u∂xj− β − γ1− γ(β − 11− γ) 1−ββ−γC(x, t)1−γβ−γD(x, t)β−1β−γ u2+u2vL[v], (x, t) ∈ G× I. (8)Theorem 4. Assume that (A1) – (A3) hold, and that there exists a non-trivial function u ∈ C1(G;R) such that u = 0 on ∂G andlimt→∞∫ tTM [u](s) ds = −∞ for any T > 0,88 Jaroslav Jarosˇ, Kusano Takasˆi and Norio YoshidawhereM [u](t) ≡∫G[n∑i,j=1Aij(x, t)∂u∂xi∂u∂xj− β − γ1− γ(β − 11− γ) 1−ββ−γC(x, t)1−γβ−γD(x, t)β−1β−γ u2]dx.Let v ∈ DL(Ω) be a solution of (1) which is nonoscillatory on Ω . Then vsatisfies the condition (4).Proof. Without loss of generality we may assume that v > 0 on G×[t0,∞)for some t0 > 0. We integrate the inequality (8) over G to obtain0 ≤ M [u](t) +∫Gu2vL[v] dx= M [u](t) +∫Gu2v∂v∂tdx, t ≥ t0or equivalently ∫Gu2∂∂tlog v dx ≥ −M [u](t), t ≥ t0.Proceeding as in the proof of Theorem 2, we see that (4) holds. Thiscompletes the proof.Corollary 3. Assume that (A1) – (A3) hold, and that there exists a non-trivial function u ∈ C1(G;R) such that u = 0 on ∂G andlimt→∞∫ tTM [u](s) ds = −∞ for any T > 0.Then every bounded solution v ∈ DL(Ω) of (1) is oscillatory on Ω.Proof. The proof follows by using the same arguments as in the proof ofCorollary 1.Corollary 4. Assume that the same hypotheses as those of Theorem 4hold. If v ∈ DL(Ω) is a solution of (1) which is nonoscillatory on Ω,then v is unbounded in Ω.Oscillatory properties of solutions of parabolic equations 89The proof is quite similar to that of Corollary 2, and will be omitted.Example 1. We consider the parabolic equation∂v∂t−(∂∂x(A0∂v∂x)+ C0v3 + C0v1/3)= 0, (9)(x, t) ∈ (−1, 1)× (0,∞),where A0 and C0 are positive constants satisfying A0 < (8/5)3−(3/4)C0.Here n = 1, A11(x, t) = A0 > 0, C(x, t) = D(x, t) = C0 > 0, β = 3,γ = 1/3, G = (−1, 1) and Ω = (−1, 1)× (0,∞). Letting u = 1− x2, we seethat u(−1) = u(1) = 0. An easy computation shows thatM [u](t) =∫ 1−1[A0(u′)2 − 4 · 3−(3/4)C0u2]dx=83(A0 − 853−(3/4)C0)< 0.Hence, it is obvious thatlimt→∞∫ tTM [u](s)ds = −∞for any T > 0. It follows from Theorem 4 that every solution v of (9) whichis nonoscillatory on Ω satisfieslimt→∞∫ 1−1(1− x2)2 log |v| dx =∞.Example 2. We consider the parabolic equation∂v∂t−(∂∂x(A0∂v∂x)+12e−4tv5 +12e(2/3)tv1/3)= 0, (10)(x, t) ∈ (0, pi)× (0,∞),where A0 is a positive constant satisfying A0 < (7/2)6−(6/7). Here n = 1,A11(x, t) = A0 > 0, C(x, t) = (1/2)e−4t, D(x, t) = (1/2)e(2/3)t, β = 5,γ = 1/3, G = (0, pi) and Ω = (0, pi) × (0,∞). Letting u = sinx, we findthat u(0) = u(pi) = 0 andM [u](t) =∫ pi0[A0(u′)2 − 726−(6/7)u2]dx=pi2(A0 − 726−(6/7))< 0.90 Jaroslav Jarosˇ, Kusano Takasˆi and Norio YoshidaHence, it is clear thatlimt→∞∫ tTM [u](s)ds = −∞for any T > 0. Then, Theorem 4 implies that every solution v of (10) whichis nonoscillatory on [0, pi]× [0,∞) satisfieslimt→∞∫ pi0(sin2 x)log |v| dx =∞.One such solution is v = et.Example 3. We consider the parabolic equation∂v∂t−(∂2v∂x2+ 5v3 + 5v1/5)= 0, (x, t) ∈ (0, pi/2)× (0,∞). (11)Here n = 1, A11(x, t) = 1, C(x, t) = D(x, t) = 5, β = 3, γ = 1/5, G =(0, pi/2) and Ω = (0, pi/2) × (0,∞). We let u = x cosx and find thatu(0) = u(pi/2) = 0. A direct calculation yieldsM [u](t) =∫ pi/20[(u′)2 − 7(52)(2/7)u2]dx=pi8((pi26+ 1)− 7(52)(2/7)(pi26− 1))< 0.Sincelimt→∞∫ tTM [u](s)ds = −∞for any T > 0, we conclude from Corollary 3 that every bounded solutionv of (11) is oscillatory on Ω.References[1] C. Y. Chan, Singular and unbounded matrix solutions for both time-dependent matrix and vector differential systems, J. Math. Anal. Appl.,87 (1982), 147–157.[2] C. Y. Chan and E. C. Young, Unboundedness of solutions and com-parison theorems for time-dependent quasilinear differential matrix in-equalities, J. Differential Equations, 14 (1973), 195–201.Oscillatory properties of solutions of parabolic equations 91[3] C. Y. Chan and E. C. Young, Singular matrix solutions for time-dependent fourth order quasilinear matrix differential inequalities, J.Differential Equations, 18 (1975), 386–392.[4] D. R. Dunninger, Sturmian theorems for parabolic inequalities, Rend.Accad. Sci. Fis. Mat. Napoli, 36 (1969), 406–410.[5] J. Jarosˇ, T. Kusano and N. Yoshida, Picone-type inequalities for nonlin-ear elliptic equations and their applications, J. Inequal. Appl., 6 (2001),387–404.[6] L. M. Kuks, Unboundedness of solutions of high-order parabolic systemsin the plane and a Sturm-type comparison theorem, Differentsial′nyeUravneniya, 14 (1978), 878–884; Differential Equations, 14 (1978), 623–627.[7] T. Kusano and M. Narita, Unboundedness of solutions of parabolicdifferential inequalities, J. Math. Anal. Appl., 57 (1977), 68–75.[8] A. McNabb, A note on the boundedness of solutions of linear parabolicequations, Proc. Amer. Math. Soc., 13 (1962), 262–265.Jaroslav JarosˇDepartment of Mathematical AnalysisFaculty of Mathematics and PhysicsComenius University842 15 Bratislava, Slovak RepublicKusano TakasˆiDepartment of Applied MathematicsFaculty of ScienceFukuoka UniversityFukuoka, 814-0180, JapanNorio YoshidaDepartment of MathematicsFaculty of ScienceToyama UniversityToyama, 930-8555, Japan(Received November 1, 2001)

Oscillatory properties of solutions of superlinear-sublinear parabolic equations via Picone-type inequalities

https://toyama.repo.nii.ac.jp/?action=repository_action_common_download&item_id=3036&item_no=1&attribute_id=18&file_no=1

Oscillatory properties of solutions of superlinear-sublinear parabolic equations via Picone-type inequalities

Abstract

Similar works

Full text

Available Versions

University of Toyama Repository