A Picone-type inequality is established for a class of superlinear-sublinear parabolic equations, and oscillatory behavior and unboundedness of solutions are investigated by using the Piconetype inequality

Yoshida Norio

application/pdfA Picone-type inequality is established for a class of superlinear-sublinear parabolic equations, and oscillatory behavior and unboundedness of solutions are investigated by using the Piconetype inequality.ArticleToyama mathematical journal，vol.30，Page 77-87departmental bulletin pape

Yoshida, Norio

University of Toyama Repository

Toyama Math. J.Vol. 30(2007), 77-87Oscillations and unboundedness of solutions ofsuperlinear-sublinear parabolic equationsvia Picone-type inequalityNorio Yoshida∗Abstract. A Picone-type inequality is established for a class ofsuperlinear-sublinear parabolic equations, and oscillatory behaviorand unboundedness of solutions are investigated by using the Picone-type inequality.In 1962, McNabb [11] established criteria for unboundedness of solutionsof linear parabolic equations on the basis of Picone identity. His resultswere extended by Dunninger [4], Kusano and Narita [10] to parabolic dif-ferential inequalities, and by Chan [1], Chan and Young [2, 3], Kobayashiand Yoshida [7], Kuks [8] to time-dependent matrix differential inequalities.All of them also contain the results about zeros of solutions or singularitiesof matrix solutions.Recently Jarosˇ, Kusano and Yoshida [5] established a Picone-type in-equality which connects a linear elliptic operator with an associated ellip-tic operator with superlinear-sublinear terms. Extending the Picone-typeinequality to parabolic equations with time-dependent coefficients, Jarosˇ,2000 Mathematics Subject Classification. 35B05, 35J70.Key words and phrases. Picone-type inequality, oscillation, unboundedness of solu-tions, superlinear-sublinear parabolic equations.∗This research was partially supported by Grant-in-Aid for Scientific Research (C)(2)(No. 16540144), The Ministry of Education, Culture, Sports, Science and Technology,Japan.78 Norio YoshidaKusano and Yoshida [5] derived the oscillatory behavior and the unbound-edness of solutions of superlinear-sublinear parabolic equations of the form∂v∂t− n∑i,j=1∂∂xi(Aij(x, t)∂v∂xj)+ C(x, t)|v|β−1v +D(x, t)|v|γ−1v = 0in a cylindrical domain Ω := G × (0,∞) ⊂ Rn+1. We note that Jarosˇ,Kusano and Yoshida [6] studied the quasilinear parabolic equation∂v∂t−[∇ · (A(x, t)|∇v|α−1∇v)+ C(x, t)|v|α−1v] = 0,where α > 0 is a constant.In this paper we deal with the quasilinear parabolic equation∂v∂t− P [v] = 0, (x, t) ∈ Ω = G× (0,∞), (1)where G is a bounded domain in Rn with piecewise smooth boundary ∂GandP [v] := ∇ · (A(x, t)|∇v|α−1∇v)+ C(x, t)|v|β−1v +D(x, t)|v|γ−1v. (2)We investigate the oscillations of solutions of (1), and the unboundednessof solutions is also obtained as corollaries.It is assumed that :(A1) A(x, t) ∈ C(Ω; (0,∞)) ;(A2) C(x, t) ∈ C(Ω; [0,∞)) and D(x, t) ∈ C(Ω; [0,∞)) ;(A3) α, β and γ are constants such that β > α and 0 < γ < α.The domain DP (Ω) of P is defined to be the set of all functions v of classC1(Ω;R) with the property that A(x, t)|∇v|α−1∇v ∈ C1(Ω;R) ∩ C(Ω;R).Definition 1. By a solution of (1) we mean a function v ∈ D(Ω) whichsatisfies (1).Definition 2. A solution v of (1) is said to be oscillatory on Ω if v hasa zero on G× [t,∞) for any t > 0. Otherwise, v is called nonoscillatory onΩ.Oscillations and unboundedness of solutions 79Associated with (2) we consider the half-linear elliptic operator p definedbyp[u] = ∇ · (a(x)|∇u|α−1∇u)+ c(x)|u|α−1u,where a(x) and c(x) satisfy the following hypothesis :(A4) a(x) ∈ C(G;R) and c(x) ∈ C(G;R).The domain Dp(G) of p is defined to be the set of all functions u of classC1(G;R) with the property that a(x)|∇u|α−1∇u ∈ C1(G;R) ∩ C(G;R).Theorem 1 (Picone-type inequality) Assume that u ∈ Dp(G), v ∈DP (Ω) and v 6= 0 in G × I, where I is any interval in (0,∞). Then wehave the Picone-type inequality∇ ·(uϕ(v)[ϕ(v)a(x)Φ(∇u)− ϕ(u)A(x, t)Φ(∇v)])≥ (a(x)−A(x, t))|∇u|α+1 + (H(x, t)− c(x))|u|α+1+A(x, t)[|∇u|α+1 + α∣∣∣uv∇v∣∣∣α+1 − (α+ 1)(∇u) · Φ(uv∇v)]+uϕ(v)(ϕ(v)p[u]− ϕ(u)P [v]), (x, t) ∈ G× I, (3)where ϕ(s) = |s|α−1s (s ∈ R), Φ(ξ) = |ξ|α−1ξ (ξ ∈ Rn) andH(x, t) =β − γα− γ(β − αα− γ)α−ββ−γC(x, t)α−γβ−γD(x, t)β−αβ−γ .Proof. The following identity was established by Kusano, Jarosˇ andYoshida [9, p.384]:∇ ·(uϕ(v)[ϕ(v)a(x)Φ(∇u)])= a(x)|∇u|α+1 − c(x)|u|α+1 + uϕ(v)(ϕ(v)p[u]). (4)It is easy to check that the Picone-type inequality which was derivedby Yoshida [12, Theorem 4.1] holds for the case where A(x) = A(x, t),80 Norio YoshidaB(x) = 0, C(x) = C(x, t), D(x) = D(x, t). Hence, we obtain the inequal-ity−∇ ·(uϕ(u)A(x, t)Φ(∇v)ϕ(v))≥ −A(x, t) |∇u|α+1 +H(x, t) |u|α+1+A(x, t)[|∇u|α+1 + α∣∣∣uv∇v∣∣∣α+1 − (α+ 1)(∇u) · Φ(uv∇v)]−uϕ(u)ϕ(v)P [v]. (5)Combining (4) with (5) yields the desired Picone-type inequality (3).The following notation will be used :V [u](t) =∫G[(a(x)−A(x, t))|∇u|α+1 + (H(x, t)− c(x))|u|α+1] dx,M [u](t) =∫G[A(x, t)|∇u|α+1 −H(x, t)|u|α+1] dx.Theorem 2. Assume that there is a nontrivial function u ∈ Dp(G) suchthatp[u] = 0 in G,u = 0 on ∂G,limt→∞∫ tTV [u](s) ds =∞ for any T > 0.If 0 < α ≤ 1, then every solution v ∈ DP (Ω) of (1) which is nonoscillatoryon Ω satisfieslimt→∞∫G|u|α+1θ(|v|) dx =∞, (6)whereθ(s) ={log s (if α = 1)s−α+1 (if 0 < α < 1).If α > 1, then every solution v ∈ DP (Ω) of (1) is oscillatory on Ω.Proof. Let 0 < α ≤ 1 and v ∈ DP (Ω) be a solution of (1) which isnonoscillatory on Ω. Then there is a number t0 > 0 such that v 6= 0 onOscillations and unboundedness of solutions 81G× [t0,∞). Integrating the Picone-type inequality (3) over G, we see that0 ≥ V [u](t)−∫Guϕ(u)ϕ(v)P [v] dx= V [u](t)−∫G|u|α+1 1|v|α−1v∂v∂tdx, t ≥ t0 (7)in view of the fact thatA(x, t)[|∇u|α+1 + α∣∣∣uv∇v∣∣∣α+1 − (α+ 1)(∇u) · Φ(uv∇v)]≥ 0(see Kusano, Jarosˇ and Yoshida [9, Lemma 2.1]). It is easy to check that1|v|α−1v∂v∂t=∂∂tlog |v| (α = 1)∂∂t(1−α+ 1 |v|−α+1)(α 6= 1)and therefore (7) impliesddt(∫G|u|α+1 log |v| dx)≥ V [u](t) (α = 1), (8)ddt(1−α+ 1∫G|u|α+1|v|−α+1 dx)≥ V [u](t) (α 6= 1) (9)for t ≥ t0. We integrate (8) and (9) over [t0, T ] to obtainΘ(T )−Θ(t0) ≥∫ Tt0V [u](s) ds (α = 1), (10)1−α+ 1(Θ(T )−Θ(t0)) ≥ ∫ Tt0V [u](s) ds (α 6= 1), (11)whereΘ(t) :=∫G|u|α+1θ(|v|) dx. (12)In case 0 < α ≤ 1, we observe, using (10) and (11), thatlimT→∞Θ(T ) =∞which is equivalent to (6).82 Norio YoshidaLet α > 1. Suppose to the contrary that there is a nonoscillatory solutionv ∈ DP (Ω) on Ω of (1). Arguing as in the proof of the first statement, wesee that (11) holds. Since −α+ 1 < 0, from (11) it follows that1α− 1Θ(t0) ≥∫ Tt0V [u](s) ds.The right hand side of the above inequality tends to ∞ as T → ∞, andtherefore a contradiction yields. This completes the proof.Corollary 1. Let 0 < α ≤ 1 and assume that there is a nontrivial func-tion u ∈ Dp(G) such thatp[u] = 0 in G,u = 0 on ∂G,limt→∞∫ tTV [u](s) ds =∞ for any T > 0.Then every bounded solution v ∈ DP (Ω) of (1) is oscillatory on Ω.Proof. Let v ∈ DP (Ω) be any bounded solution of (1). We easily seethat θ(|v|) is bounded from above, and so is ∫G |u|α+1θ(|v|)dx. Then (6)does not hold, hence Theorem 2 implies that the (bounded) solution v isoscillatory on Ω.Corollary 2. Let 0 < α ≤ 1 and assume that the same hypotheses asthose of Theorem 2 hold. If v ∈ DP (Ω) is a solution of (1) which is nonoscil-latory on Ω, then v is unbounded in Ω.Proof. Since v is nonoscillatory on Ω, it follows from Theorem 2 that vsatisfies the condition (6). Hence, |v| cannot be bounded from above in Ω,that is, v is unbounded in Ω.Theorem 3. Assume that there is a nontrivial function u ∈ C1(G;R)such that u = 0 on ∂G andlimt→∞∫ tTM [u](s) ds = −∞ for any T > 0. (13)If 0 < α ≤ 1, then every solution v ∈ DP (Ω) of (1) which is nonoscillatoryon Ω satisfies (6). If α > 1, then every solution v ∈ DP (Ω) of (1) isoscillatory on Ω.Oscillations and unboundedness of solutions 83Proof. Let 0 < α ≤ 1 and assume that v ∈ DP (Ω) is a solution of (1)which is nonoscillatory on Ω. Then v 6= 0 on G × [t0,∞) for some t0 > 0.In Theorem 2 we used the Picone-type inequality (3). Integrating (5) overG instead of (3), we obtain0 ≤ M [u](t) +∫G|u|α+1 1|v|α−1vP [v] dx= M [u](t) +∫G|u|α+1 1|v|α−1v∂v∂tdx, t ≥ t0.Proceeding as in the proof of Theorem 2, we observe thatΘ(T )−Θ(t0) ≥ −∫ Tt0M [u](s) ds (α = 1),1−α+ 1(Θ(T )−Θ(t0)) ≥ −∫ Tt0M [u](s) ds (α 6= 1),where Θ(t) is given by (12). Using the same arguments as in the proofof Theorem 2, we find that v satisfies (6). The case where α > 1 canbe handled by an argument similar to that of Theorem 2. The proof iscomplete.Corollary 3. Let 0 < α ≤ 1 and assume that there is a nontrivial func-tion u ∈ C1(G;R) satisfying (13) and the boundary condition u = 0 on ∂G.Then every bounded solution v ∈ DP (Ω) of (1) is oscillatory on Ω.Corollary 4. Let 0 < α ≤ 1 and assume that there is a nontrivial func-tion u ∈ C1(G;R) satisfying (13) and the boundary condition u = 0 on ∂G.If v ∈ DP (Ω) is a solution of (1) which is nonoscillatory on Ω, then v isunbounded in Ω.Corollaries 3 and 4 follow from Theorem 3, and the proofs of them arequite similar to those of Corollaries 1 and 2, respectively, and will be omit-ted.Example 1. We consider the quasilinear parabolic equation∂v∂t−[∂∂x(A0∣∣∣∣∂v∂x∣∣∣∣α−1 ∂v∂x)+ C0v3 + C0v1/3]= 0,(x, t) ∈ (−1, 1)× (0,∞), (14)84 Norio Yoshidawhere A0 and C0 are positive constants. Here n = 1, A(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,∞). We consider the two cases where α = 2 or α = 1/2. Firstwe treat the case where α = 2. Choosing u = 1 − x2, we observe thatu(−1) = u(1) = 0. It is easily verified thatH(x, t) = H0 =85(53)3/8C0.An easy calculation yieldsM [u](t) =∫ 1−1[A0|u′(x)|3 −H0|u(x)|3]dx= 4A0 − 3235H0.If A0 < (8/35)H0, then the condition (13) is satisfied, and therefore The-orem 3 implies that every solution v of (14) with α = 2 is oscillatory onΩ.Next we deal with the case where α = 1/2. Choosing u = 1−x2, we findthat u(−1) = u(1) = 0 andM [u](t) =∫ 1−1[A0|u′(x)|3/2 −H0|u(x)|3/2]dx=85√2A0 − 38piH0.If A0 < (15/128)√2piH0, we see that the condition (13) is satisfied, andtherefore Theorem 3 implies that every solution v of (14) with α = 1/2which is nonoscillatory on Ω satisfieslimt→∞∫ 1−1(1− x2)3/2 |v|1/2dx =∞.Example 2. We consider the quasilinear parabolic equation∂v∂t−[∂∂x(A0∣∣∣∣∂v∂x∣∣∣∣ ∂v∂x)+12e−4tv5 +12e(2/3)tv1/3]= 0, (15)(x, t) ∈ (0, pi)× (0,∞),Oscillations and unboundedness of solutions 85where A0 is a positive constant. Here n = 1, α = 2, β = 5, γ = 1/3,A(x, t) = A0 > 0, C(x, t) = (1/2)e−4t, D(x, t) = (1/2)e(2/3)t, G = (0, pi)and Ω = (0, pi)× (0,∞). Choosing u = sinx, we find that u(0) = u(pi) = 0,H(x, t) = H0(t) =75(59)9/14e−tandM [u](t) =∫ pi0[A0|u′(x)|3 −H(x, t)|u(x)|3]dx= 2A0∫ pi/20cos3 x dx− 2H0(t)∫ pi/20sin3 x dx=43(A0 −H0(t)).Hence, the condition (13) is violated. Then, there exists a nonoscillatorysolution v = et of (15).Example 3. We consider the quasilinear parabolic equation∂v∂t−[∂∂x(A0∣∣∣∣∂v∂x∣∣∣∣−1/2 ∂v∂x)+ C0v3 + C0v1/5]= 0, (16)(x, t) ∈ (0, pi/2)× (0,∞),where A0 and C0 are positive constants. Here n = 1, α = 1/2, β = 3,γ = 1/5, A(x, t) = A0 > 0, C(x, t) = D(x, t) = C0 > 0, G = (0, pi/2) andΩ = (0, pi/2)× (0,∞). Letting u = x cosx, we see that u(0) = u(pi/2) = 0andM [u](t) =∫ pi/20[A0|u′(x)|3/2 −H0|u(x)|3/2]dx,whereH0 =283(325)25/28C0.If C0 is sufficiently large, then M [u](t) is a negative constant, and there-fore the condition (13) is satisfied. From Corollary 3 it follows that everybounded solution v of (16) is oscillatory on Ω.86 Norio YoshidaReferences[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.[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, Oscillatory properties of solutionsof superlinear-sublinear parabolic equations via Picone-type inequali-ties, Math. J. Toyama Univ., 24 (2001), 83–91.[6] J. Jarosˇ, T. Kusano and N. Yoshida, Oscillation properties of solutionsof a class of nonlinear parabolic equations, J. Comput. Appl. Math.,146 (2002), 277–284.[7] K. Kobayashi and N. Yoshida, Unboundedness of solutions of time-dependent differential systems of parabolic type, Math. J. ToyamaUniv., 25 (2002), 65–75.[8] L. M. Kuks, Unboundedness of solutions of high-order parabolic sys-tems in the plane and a Sturm-type comparison theorem, Differ.Uravn., 14 (1978), 878–884; Differ. Equ., 14 (1978), 623–627.[9] T. Kusano, J. Jarosˇ and N. Yoshida, A Picone-type identity and Stur-mian comparison and oscillation theorems for a class of half-linear par-tial differential equations of second order, Nonlinear Anal., 40 (2000),381–395.Oscillations and unboundedness of solutions 87[10] T. Kusano and M. Narita, Unboundedness of solutions of parabolicdifferential inequalities, J. Math. Anal. Appl., 57 (1977), 68–75.[11] A. McNabb, A note on the boundedness of solutions of linear parabolicequations, Proc. Amer. Math. Soc., 13 (1962), 262–265.[12] N. Yoshida, Picone-type inequalities for a class of quasilinear ellipticequations and their applications, Proceedings of the Conference onDifferential & Difference Equations and Applications (Florida, 2005),pp. 1177–1185, New York, 2006.Norio YoshidaDepartment of MathematicsUniversity of ToyamaToyama, 930-8555Japan(Received August 30, 2007)

Oscillations and unboundedness of solutions of superlinear-sublinear parabolic equations via Picone-type inequality

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Oscillations and unboundedness of solutions of superlinear-sublinear parabolic equations via Picone-type inequality

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