LIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS

Transcription

1 Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp ISSN: URL: or ftp ejde.math.txstate.edu LIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS FUQIN SUN Abstract. In this article, we study the higher-order semilinear parabolic equation u t + ( ) m u = u p, (t, x) R 1 + RN, u(, x) = u (x), x R N. Using the test function method, we derive the blow-up critical exponent. And then based on integral inequalities, we estimate the life span of blow-up solutions. 1. Introduction This article concerns the cauchy problem for the higher-order semilinear parabolic equation u t + ( ) m u = u p, (t, x) R 1 + R N, (1.1) u(, x) = u (x), x R N, where m, p > 1. Higher-order semilinear and quasilinear heat equations appear in numerous applications such as thin film theory, flame propagation, bi-stable phase transition and higher-order diffusion. For examples of these mathematical models, we refer the reader to the monograph [9]. For studies of higher-order heat equations we refer also to [1, 2, 4, 5, 6, 1] and the references therein. In [6], under the assumption that u L 1 (R N ) L (R N ), u and R N u (x)dx, (1.2) Galaktionov and Pohozaev studied the Fujita critical exponent of problem (1.1) and showed that p F = 1 + /N. The critical exponents p F is calculated from both sides: (i) blow-up of any solutions with (1.2) for 1 < p p F and (ii) global existence of small solutions for p > p F. 2 Mathematics Subject Classification. 35K3, 35K65. Key words and phrases. Higher-order parabolic equation; critical exponent; life span; test function method. c 21 Texas State University - San Marcos. Submitted August 17, 29. Published January 27, 21. Supported by grant from the National Natural Science Foundation of China. 1

2 2 F. SUN EJDE-21/17 Egorov et al [5] studied the asymptotic behavior of global solutions with suitable initial data in the supercritical Fujita range p > p F by constructing self-similar solutions of higher-order parabolic operators and through a stability analysis of the autonomous dynamical system. For other studies of the problem, we refer to [4] where global non-existence was proved for p (1, p F ] by using the test function approach, and [1] where a general situation was discussed with nonlinear function h(u) in place of u p. In a recent paper [1], we discussed the system u t + ( ) m u = v p, (t, x) R 1 + R N, v t + ( ) m v = u q, (t, x) R 1 + R N, u(, x) = u (x), v(, x) = v (x), x R N. (1.3) It is proved that if N/( > max { 1+p pq 1, pq 1} 1+q then solutions of (1.3) with small initial data exist globally in time. Moreover the decay estimates u(t) C(1 + t) σ1 and v(t) C(1 + t) σ2 with σ 1 > and σ 2 > are also satisfied. On the other hand, under the assumption that u (x)dx >, v (x)dx >, R N R N if N/() max { 1+p pq 1, 1+q pq 1} then every solution of (1.3) blows up in finite time. In our present work, exploiting the test function method, we shall give the life span of blow-up solution for some special initial data. The main idea comes from [7] for discussing cauchy problem of the second order equation ρ(x)u t u m = h(x, t)u 1+p, (t, x) R 1 + R N, u(, x) = u (x), x R N. (1.4) Using the test function method, the author gave the blow-up type critical exponent and the estimates for life span [, T ) like that in [8]. For the construction of a test function, the author mainly based on the eigenfunction Φ corresponding to the principle eigenvalue λ 1 of the Dirichlet problem on unit ball B 1, w(x) = λ 1 w(x), x B 1, w(x) =, x B 1. However, for the operator ( ) m, the eigenfunction Φ corresponding to the principal eigenvalue λ 1 of the Dirichlet problem may change sign (see [3]). We will use a non-negative smooth function Φ constructed in [1] and [6]. The organization of this paper is as follows. In section 2, by the test function method, we derive some integral inequalities and reacquire the Fujita critical exponent p F obtained in the paper [6]. Section 3 is for the estimate of life span of blow-up solution. 2. Fujita critical exponent In this section, we shall use the test function method to derive the Fujita critical exponent and some useful inequalities. From the reference [6], we know that if u L 1 (R N ) L (R N ), then the solution u(t, ) C 1 ([, T ]; L 1 (R N ) L (R N )) for some T >. Therefore, without loss of generality, we may consider u (x) concentrated around the origin and bounded below by a positive constant in some

3 EJDE-21/17 LIFE SPAN OF BLOW-UP SOLUTIONS 3 neighborhood of origin. Further, u (x) as x. With these choices, the solution u and its spatial derivatives vanish as x for t >. First we construct a test function. For this aim, we shall use a non-negative smooth function Φ which was constructed in the papers [1] and [6]. Let Φ(x) = Φ( x ) >, Φ() = 1; < Φ(r) 1 for r >, where Φ(r) is decreasing and Φ(r) as r sufficiently fast. Moreover, there exists a constant λ 1 > such that and such that m Φ λ 1 Φ, x R N, (2.1) Φ 1 = Φ(x)dx = 1. R N This can be done by letting Φ(r) = e rν for r 1 with ν (, 1], and then extending Φ to [, ) by a smooth approximation. Take θ > p/(p 1), and define, t > T, φ(t) = (1 (t S)/(T S)) θ, t T, 1, t < S, where S < T. Now set ξ(t, x) = φ(t/r )Φ(x/R), R >. Suppose that u exists in [, t ) R N. For T R < t, multiply both sides of equation (1.1) by ξ and integrate over [, T R ) R N by parts to obtain T R Denote u p ξdxdt + u (x)ξ(, x)dx R N R N I(S, T ) = T R SR T R R N u { ξ t + m ξ }dxdt. u p φ(t/r )Φ(x/R)dxdt, R N J = u (x)φ(x/r)dx. R N (2.2) We now estimate I(, T ) + J. Using the Hölder inequality, since φ (t) = except on (S, T ), we obtain T R T R u ξ t dxdt = u φ(t/r ) 1/p φ (t/r ) R N SR R N φ(t/r ) 1/p Φ(x/R)R dxdt I(S, T ) 1/p R ( T R φ (t/r ) p/(p 1) SR R N (p 1)/p. φ(t/r ) Φ(x/R)dxdt) 1/(p 1) (2.3)

4 4 F. SUN EJDE-21/17 Since m x Φ(x/R) = R m y Φ(y) for y = x/r, using the Hölder inequality and (2.1) we have T R u m ξ dxdt R N T R = u φ(t/r ) m Φ(x/R) dxdt R N T R = R u φ(t/r ) m x/r Φ(x/R) dxdt (2.4) R N T R λ 1 R u φ(t/r )Φ(x/R)dxdt R N I(, T ) 1/p λ 1 R ( T R ) (p 1)/p. φ(t/r )Φ(x/R)dxdt R N Making the change of variables τ = t/r and η = x/r, from (2.2), (2.3) and (2.4) we deduce that I(, T ) + J I(S, T ) 1/p R s( T R ) (p 1)/p φ (τ) p/(p 1) φ(τ) 1/(p 1) Φ(η)dηdτ (2.5) SR R N + I(, T ) 1/p λ 1 R s( T ) (p 1)/p, φ(τ)φ(η)dηdτ R N where s = + ( + N)(p 1)/p. Set ( T A(S, T ) = φ (τ) p/(p 1) φ(τ) 1/(p 1) Φ(η)dηdτ S R N ( T ) (p 1)/p. B(T ) = φ(τ)φ(η)dηdτ R N Thus (2.5) can be simply written as ) (p 1)/p, I(, T ) + J R s [I(S, T ) 1/p A(S, T ) + λ 1 I(, T ) 1/p B(T )]. (2.6) We have the following result: Theorem 2.1 (Fujita critical exponent). If R N u (x)dx, u (x) and s, that is to say p p c = 1 + /N, then (1.1) has no global solution. Proof. By slightly shifting the origin in time, we may assume Let u be a global solution with u satisfying (2.7), then u p dxdt >. R N R N u (x)dx >. (2.7)

5 EJDE-21/17 LIFE SPAN OF BLOW-UP SOLUTIONS 5 Suppose s <. Letting R tend to infinity in (2.6) to obtain u p dxdt + u (x)dx =. R N R N Hence u, a contradiction. Suppose s =. We first show J for all R >. In fact, from the assumptions on initial datum, there exists ε > such that u (x) δ > for x ε. Set J = u (x)φ(x/r)dx + u (x)φ(x/r)dx x ε x >ε > δ Φ(x/R)dx + u (x)φ(x/r)dx x ε x >ε = δr N Φ(η)dη + u (x)φ(x/r)dx η ε /R x >ε u (x)φ(x/r)dx. x >ε By the choice of Φ, we have lim R x >ε u (x)φ(x/r)dx =. And so there exists R > such that J for all < R < R. On the other hand, there exists M > such that u (x)dx > u (x) dx. x R M x >R M In addition, by a slight modification of Φ, we may set Φ(x) 1 in {x : x M}. Note that since Φ 1 we have, for R R, J = u (x)φ(x/r)dx + u (x)φ(x/r)dx x R M x R M x R M u (x)dx u (x)dx x >R M x >R M x >R M u (x) Φ(x/R)dx u (x) dx >. Now we are in the position to complete the proof of case s =. Since θ(t S) 1/p [ A(S, T ) = [θ 1/(p 1)], B(T ) = S + T S (p 1)/p θ + 1 ] (p 1)/p, we may choose S small and θ large, T S bounded, such that ( B(T ) u (x)dx/ [2λ ) 1/p ] 1 u p dxdt. (2.8) R N R N Moreover, note that J, from (2.6) we get that I(, T ) is uniformly bounded for all R >. Then, keeping T S bounded, lim I(S, T R )1/p A(S, T ) =. (2.9) Letting R, (2.6) (2.9) give u p dxdt + 1 u (x)dx =, R 2 N R N

6 6 F. SUN EJDE-21/17 which also implies u. Let σ be an arbitrary positive number. For x [, ) and < ω < 1, define Ψ(ω; σ) := max x (σxω x). It is easy to check that Ψ(ω; σ) = (1 ω)ω ω 1 ω σ 1 1 ω. Set We have the following result. A(T ) = A(, T ), S(T ) = A(T ) + λ 1 B(T ). Theorem 2.2. If u is a solution of (1.1) defined on [, t ) R N. Then, for R > and τ t R, we have ( 1 u (x)φ(x/r)dx Ψ R p ; S(T )Rs). (2.1) N Moreover, if u is a global solution of (1.1), then lim sup R ŝ R u (x)φ(x/r)dx λ 1/(p 1) 1, R N (2.11) where ŝ = sp/(p 1). Proof. Denote I(T ) = I(, T ). Firstly, by the definition of Ψ, from (2.6) we know that J I(T ) 1/p S(T )R s I(T ) Ψ( 1 p ; S(T )Rs). This is exactly (2.1). By means of (2.1), we deduce that ( 1 u (x)φ(x/r)dx Ψ ; S(T )Rs) R p N = (1 1/p)(1/p) 1/p 1 1/p [S(T )R s ] 1 1 1/p = (p 1)p p/(1 p) R sp/(p 1) S(T ) p p 1, (2.12) which leads to lim sup R ŝ u (x)φ(x/r)dx (p 1)p p/(1 p) [inf S(T )] p p 1. (2.13) R R N T To estimate S(T ), we need estimate A(T ) and B(T ) respectively. Denote We obtain Since a p = θ [θ 1/(p 1)], b λ 1 (p 1)/p p = (θ + 1). (p 1)/p S(T ) = a p T 1/p + b p T (p 1)/p. min S(T ) = p[a p/(p 1)] (p 1)/p b 1/p p T = p(p 1) (p 1)/p λ 1/p 1 θ (p 1)/p [θ 1/(p 1)] (p 1)2 /p 2 (1 + θ) (p 1)/p2, we have lim p(t ) = p(p 1) (p 1)/p λ 1/p 1. θ T (2.14) Combining (2.13) and (2.14), we obtain (2.11). The proof is complete.

7 EJDE-21/17 LIFE SPAN OF BLOW-UP SOLUTIONS 7 3. Life span of blow-up solutions In this section, we shall estimate the life span of the blow-up solution with some special initial datum. To this aim, we assume that u satisfies (H) There exist positive constants C, L such that { δ, x ε, u (x) C x κ, x > ε, where δ and ε are as in the proof of Theorem 2.1, and N < κ < /(p 1) if p < 1 + /N; < κ < N if p = 1 + /N. Now we state the main result. Theorem 3.1. Let (H) be fulfilled and u ε be the solution of (1.1) with initial data u ε (, x) = εu (x), where ε >. Denote [, T ε ) be the life span of u ε. Then there exists a positive constant C such that T ε Cε 1/ ˆβ, where ˆβ = κ 1 p 1 <. Remark 3.2. When p = 1 + /N, note that ˆβ = (κ N)/(). Proof. Choose R such that R R >. By the definition of J and the assumptions of initial data, we have J = ε u (x)φ(x/r)dx R N ε u (x)φ(x/r)dx x >ε = εr N u (Rη)Φ(η)dη η >ε /R (3.1) εc R N κ η κ Φ(η)dη η >ε /R εc R N κ η >ε /R η κ Φ(η)dη = CR N κ. Using (2.12), we know from (3.1) that, for < τ < T ε, ε R κ N C 1 (p 1)p p/(1 p) [R s S(T )] p/(p 1) = C 1 (p 1)p p/(1 p) H(τ, R), (3.2) where H(τ, R) = R κ N [S(τR )R s ] p/(p 1). We write H(τ, R) = [a p τ 1/p R α1 + b p τ (p 1)/p R α2 ] p/(p 1), where α 1 = (p 1)κ/p, α 2 = (p 1)κ/p. The choice of κ implies α 1, α 2 >. Now we derive some estimates on H(τ, R). If we can find a function G(τ) such that H(τ, R) G(τ), τ >,

8 8 F. SUN EJDE-21/17 and for each value of R R there exists a value of τ R such that H(τ R, R) = G(τ R ), then (3.2) holds for all R R if and only if Set Then ε C 1 (p 1)p p/(1 p) G(τ). (3.3) y = R α1+α2 = R, β 1 = α 2 /(α 1 + α 2 ) = α 2 /(). H(τ, R) = τ 1/(p 1) h(τ, y) p/(p 1) with h(τ, y) = a p y 1 β1 + b p y β1 τ. Denote where σ = a p b 1 p (1 β 1 )β1 1 y, G(τ) = τ 1/(p 1) g(τ) p/(p 1), g(τ) = [a p y 1 β1 σ β1 1 + b p y β1 σ β1 ]τ 1 β1. It is easy to check that < β 1 < 1. Then, ζ = g(τ) ia a concave curve. Furthermore, ζ = h(τ, y) is a tangent line of ζ = g(τ) at the point of (σ, g(σ)). Therefore, we get that h(τ, y) g(τ), for all τ >. Hence H(τ, R) G(τ), for all τ >. Moreover, H(τ, R τ ) = G(τ) with τ R = a p b 1 p (1 β 1 )β1 1 R. By computations, for some positive constant C, where G(τ) = τ 1/(p 1) g(τ) p/(p 1) = C 1 τ ˆβ. (3.4) ˆβ = κ 1 p 1. The choice of κ implies that ˆβ <. Combining (3.3) and (3.4), we find that ε Kτ ˆβ for some K >. From (3.5), it follows that for some C >. The proof is complete. 1/ ˆβ τ Cε References (3.5) [1] M. Chaves, V. A. Galaktionov, Regional blow-up for a higher-order semilinear parabolic equation, Europ J. Appl. Math., 12 (21), [2] S. B. Cui, Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Analysis TMA, 43 (21), [3] Yu. V. Egorov, V. A. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser Verlag, Basel Boston-Berlin, [4] Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev, S. I. Pohozaev, On the necessary conditions of global existence to a qusilinear inequality in the half-space, C. R. Acad. Aci. Paris Sér. I, 33 (2), [5] Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev, S. I. Pohozaev, Global solutions of higher-order semilinear parabolic equations in the supercritical range, Adv. Diff. Equ., 9 (24), [6] V. A.Galaktionov, S. I. Pohozaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J., 51 (22), [7] H. J. Kuiper, Life span of nonnegtive solutions to certain qusilinear parabolic cauchy problems, Electronic J. of Differential equations., 23 (23), no. 66, 1 11.

9 EJDE-21/17 LIFE SPAN OF BLOW-UP SOLUTIONS 9 [8] T. Y. Lee, W. M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333(1)(1992), [9] L. A. Peletier, W. C. Troy, Spacial Patterns: Higher Order Models in Physics and Mechanics, Birkhäuser, Boston-Berlin, 21. [1] Y. H. P. Pang, F. Q. Sun, M. X. Wang, Existence and non-existence of global solutions for a higher-order semilinear parabolic system, Indiana Univ. Math. J., 55 (26), no. 3, Fuqin Sun School of Science, Tianjin University of Technology and Education, Tianjin 3222, China address: sfqwell@163.com