Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 451619, 11 pages doi:10.1155/2010/451619 Research Article Existence and Nonexistence of Global Solutions of the Quasilinear Parabolic Equations with Inhomogeneous Terms Yasumaro Kobayashi Faculty of Urban Liberal Arts, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan Correspondence should be addressed to Yasumaro Kobayashi, yasumaro@hkg.odn.ne.jp Received 20 April 2010; Accepted 14 October 2010 Academic Editor: Abdelkader Boucherif Copyright q 2010 Yasumaro Kobayashi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the quasilinear parabolic equation with inhomogeneous term u t Δu m x σ u p fx, ux, 0u 0 x,where0 ≤ fx, u 0 x ∈ CR N , m>0, p>max{1,m},andσ>−2, x :|x| 2  1 1/2 . In this paper, we investigate the critical exponents of this equation. 1. Introduction We consider the quasi-linear parabolic equation with inhomogeneous term u t Δu m  x σ u p  f  x   x ∈ R N ,t>0  , u  x, 0   u 0  x   x ∈ R N  , 1.1 where 0 ≤ fx, u 0 x ∈ CR N , m>0, p>max{1,m},andσ>−2, x :|x| 2  1 1/2 . For the solution ux, t of 1.1,letT ∗ > 0 be the maximal existence time, that is, T ∗ : sup  T>0; sup t∈  0,T   u  ·,t   ∞ < ∞  . 1.2 If T ∗  ∞, we say that ux, t is a global solution; if T ∗ < ∞, we say that ux, t blows up in finite time. 2 Advances in Difference Equations For quasi-linear parabolic equations, the authors of 1–5 and so on. study the homogeneous equations i.e., fx ≡ 0in1.1. Baras and Kersner 1 proved that 1.1 with m  1andfx ≡ 0 has a global solution, two constants c 1 and c 2 depending on N and p exist such that lim inf r →∞  r −2/p−1  |x|<r dx  x  σ/p−1  ≥ c 1  u 0 dx, lim inf |x|→∞  x  σ2 u 0  x  p−1 ≤ c 2 . 1.3 Mochizuki and Mukai 2 and Qi 4 study the case m>0, σ  0, Pinsky 3 studies the case m  1, σ>−2, and Suzuki 5 studies the case m ≥ 1, −∞ <σ<∞. The following two results are proved by them: 1 if p ≤ p ∗ m,σ , then every nontrivial solution ux, t of 1.1 blows up in finite time; 2 if p>p ∗ m,σ , then 1.1 has a global solution for some initial value u 0 x, where p ∗ m,σ  m 2  σ/N for N ≥ 2, σ>−2andforN  1, σ>−1, p ∗ m,σ  m  1forN  1, σ ≤−1. This p ∗ m,σ is called the critical exponent. On the other hand, 6–9 and so on. study the inhomogeneous equations i.e., fx / ≡ 0 in 1.1.Bandleetal.6 study the case m  1, σ  0, and Zeng 8 and Zhang 9 study the case σ  0. In this paper, we investigate the critical exponents of 1.1 in the case fx / ≡ 0. Our results are as follows. Theorem 1.1. Suppose that N ≥ 3, σ>−2, m>N − 2/N  σ, and p>max{1,m}.Put p ∗ m,σ : m  N  σ  N − 2 . 1.4 a If p ≤ p ∗ m,σ , then every nontrivial solution ux, t of 1.1 blows up in finite time. b If p>p ∗ m,σ , u 0 x ≤ C 1 x −2σ/p−m , and fx ≤ C 2 x −m2σ/p−m−4 ,then1.1 has a global solution for some constants C 1 and C 2 . Theorem 1.2. Suppose that N  1, 2, σ ≥−2, m>0, and p>max{1,m}. Then every nontrivial solution ux, t of 1.1 blows up in finite time. Remark 1.3. Theorems 1.1 and 1.2 are the extension of the results of 8.Ifweputσ  0in these theorems, the same results as Theorem 1 in 8 are obtained. We will prove Theorem 1.1a and b in Sections 3 and 4, respectively. The proof of Theorem 1.2 is included in the proof of Theorem 1.1a. In the following, R and T are two given positive real numbers greater than 1. C is a positive constant independent of R and T, and its value may change from line to line. 2. Preliminaries In this section, we first give the definition of a solution for Problem 1.1 and then cite the comparison theorem and a known result. Advances in Difference Equations 3 Definition 2.1. A continuous function u  ux, t is called a solution of Problem 1.1 in Q T ≡ R N × 0,T if the following holds: i ∇ x u m ∈ L 2 loc R N ; ii for any bounded domain D ⊂ R N and for all ψ ∈ C 2 D × 0,T and vanishing on ∂D × 0,T,  τ 0  D  u∂ t ψ −∇u m ∇ψ   x  σ u p ψ  fψ  dx dt   D u  x, ·  ψ  x, ·    τ 0 dx, 2.1 for all τ ∈ 0,T. Lemma 2.2 the comparison theorem. Let u, v ∈ C0, T; L 2 loc Ω, ∇u m , ∇v m ∈ L 2 0, T; L 2 loc Ω, and satisfy u t − Δu m ≤ v t − Δv m ,  x, t  ∈ Ω T , u ≤ v,  x, t  ∈ ∂Ω T . 2.2 Then u ≤ v for all x, t ∈ Ω T ,whereΩ is a bounded domain in R N with smooth boundary ∂Ω or ΩR N and Ω T Ω× 0,T. Lemma 2.3 the monotonicity property. Let u x be a nonnegative sub-solution to the stationary problems of Problem 1.1. Then the positive solution ux, t with initial data u x is monotone increasing to t. 3. Proof of Theorem 1.1(a) We first consider the following problem: u t Δu m   x  σ u p  f  x   x ∈ R N ,t>0  , u  x, 0   0  x ∈ R N  . 3.1 It is clear that the positive solution of Problem 3.1 is a sub-solution of Problem 1.1. If every positive solution of Problem 3.1 blows up in finite time, then, by Lemma 2.2, every positive solution of Problem 1.1 also blows up in finite time. Therefore, we only need to consider Problem 3.1. The stationary problem of Problem 3.1 is as follows: Δu m   x  σ u p  f  x   0  x ∈ R N  . 3.2 It is obvious that 0 is a sub-solution of Problem 3.2 and does not satisfy Problem 3.2.Thus, by making use of Lemmas 2.2 and 2.3, the positive solution of Problem 3.1 is monotone increasing to t. 4 Advances in Difference Equations We argue by contradiction. Assume that Problem 3.1 has a global positive solution for p ≤ p ∗ m,σ . Let ϕr and ηt be two functions in C ∞ 0, ∞, and satisfy i 0 ≤ ϕr ≤ 1in0, ∞; ϕr ≡ 1in0, 1, ϕr ≡ 0in2, ∞; −C ≤ ϕ  r ≤ 0, |ϕr|≤C; ii 0 ≤ ηt ≤ 1in0, ∞; ηt ≡ 1in0, 1, ηt ≡ 0in2, ∞; −C ≤ η  t ≤ 0. For R>1andT>1, define Q R,T ≡ B 2R × 0, 4T,andletΨr, tϕ R rη T t be a cut-off function, where ϕ R rϕr/R, η T tηt/2T. It is easy to check that − C R ≤ dϕ R  r  dr ≤ 0,      d 2 ϕ R  r  dr 2      ≤ C R 2 , − C 2T ≤ dη T  t  dt ≤ 0. 3.3 Let I R   Q R,T x σ u p Ψ s dxdt, 3.4 where s>1 is a positive number to be determined. Then I R   Q R,T  −u∂ t Ψ s  ∇u m ∇Ψ s − fΨ s  dx dt   B 2R ux, ·Ψr, · s   4T 0 dx  −  Q R,T uϕ s R dη s T dt dx dt   Q R,T ∇u m η s T ∇ϕ s R dx dt −  Q R,T fΨ s dx dt   B 2R u  x, ·  ϕ R  r  s η T  ·  s   4T 0 dx  −  Q R,T uϕ s R dη s T dt dx dt −  Q R,T u m η s T Δϕ s R dx dt −  Q R,T fΨ s dx dt   4T 0  |x|2R u m η s T ∂ϕ s R ∂ν dS dt. 3.5 Since  R N fxdx > 0, there exist δ>0andR 0 > 1 such that  B R fxdx ≥ δ for R>R 0 :  Q R,T fΨ s dx dt   4T 0 η s T  B 2R fϕ s R dx dt ≥  2T T  B R fdxdt ≥ δT. 3.6 Hence, by the definition of ϕ R and η T , we have I R ≤−  4T 2T  B 2R uϕ s R dη s T dt dx dt −  4T 0  B 2R \B R u m η s T Δϕ s R dx dt − δT. 3.7 Advances in Difference Equations 5 Since Δϕ s R  sϕ s−1 R Δϕ R  ss − 1ϕ s−2 R |∇ϕ R | 2 and Δϕ R  r   d 2 ϕ R  r  dr 2  N − 1 r dϕ R  r  dr ,   ∇ϕ R   2   dϕ R r dr  2 , 3.8 we obtain from 3.3 that   Δϕ s R   ≤ sϕ s−1 R  C R 2  N − 1 R · C R   s  s − 1  ϕ s−2 R  C R  2 ≤ C R 2 ϕ s−2 R 3.9 in B 2R \ B R and dη s T dt  sη s−1 T dη T dt ≥−sη s−1 T C 2T ≥− C T η s−1 T 3.10 in 2T, 4T.Thus,3.7 becomes I R ≤ C T  4T 2T  B 2R uΨ s−1 dx dt  C R 2  4T 0  B 2R \B R u m Ψ s−2 dx dt − δT. 3.11 Let s be large enough such that s − 1p ≥ s and s − 2p/m ≥ s,andletA σ R be as follows: A σ  R   ⎧ ⎨ ⎩ R N−σ/p−1  σ<N  p − 1  , log  R  1   σ ≥ N  p − 1  . 3.12 Then, by making use of Young’s inequality, we have C T  4T 2T  B 2R uΨ s−1 dx dt ≤  4T 2T  B 2R  1 4 p p  x  σ u p Ψ s−1p  4 q q  x  −σq/p C q T −q  dx dt ≤ 1 4  4T 0  B 2R  x  σ u p Ψ s dx dt  CT −p/p−1  4T 2T  B 2R  x  −σ/p−1 dx dt ≤ 1 4 I R  CT 1−p/p−1 A σ  R  , 3.13 6 Advances in Difference Equations where 1/p  1/q  1and C R 2  4T 0  B 2R \B R u m Ψ s−2 dx dt ≤  4T 0  B 2R \B R  1 4 p  p   x  σ u mp  Ψ s−2p   4 q  q  x −σq  /p  C q  R −2q   dx dt ≤ 1 4  4T 0  B 2R  x  σ u p Ψ s dx dt  CR −2p/p−m  4T 0  B 2R \B R  x  −mσ/p−m dx dt ≤ 1 4 I R  CTR −2p/p−m R N−mσ/p−m , 3.14 where p   p/m,1/p   1/q   1. Thus, 3.11 becomes I R ≤ 1 2 I R  T  CT −p/p−1 A σ  R   CR N−2pmσ/p−m − δ  . 3.15 For N ≥ 3, since σ>−2, 1/p   1/q   1, and max{1,m} <p≤ mN  σ/N − 2, we have N − 2p  mσ p − m   N − 2  p −  N  σ  m p − m ≤ 0. 3.16 For N  2, since σ ≥−2, m>0, and p>max{1,m}, we have 2 − 2p  mσ p − m  −  2  σ  m p − m ≤ 0. 3.17 For N  1, since σ ≥−2,m>0, and p>max{1,m}, we have 1 − 2p  mσ p − m  −p −  1  σ  m p − m < −  2  σ  m p − m ≤ 0. 3.18 Let T ≥ A σ R p−1/p such that T −p/p−1 A σ R ≤ 1, then I R ≤ CT, 3.19 that is,  4T 0  B 2R  x  σ u p Ψ s dx dt ≤ CT. 3.20 Thus  2T T  B R  x  σ u p dx dt ≤ CT. 3.21 Advances in Difference Equations 7 By the integral mean value theorem, there exists t 1 ∈ T, 2T such that  2T T  B R  x  σ u p dx dt  T  B R  x  σ u  x, t 1  p dx ≤ CT, 3.22 that is,  B R  x  σ u  x, t 1  p dx ≤ C. 3.23 Since T is a large positive number and a random selection, and ux, t is monotone increasing to t, there exists a positive number TR > 1 for any fixed R>R 0 such that, for all t>TR,  B R  x  σ u  x, t  p dx ≤ C. 3.24 By the monotone increasing property of ux, t,  B R x σ ux, t p dx also is increasing to t.This, combined with 3.24, yields that the limit I ∞ R exists such that I ∞ R ≡ lim t →∞  B R  x  σ u  x, t  p dx ≤ C. 3.25 Since ux, t is nonnegative, I ∞ R is monotone increasing to R. This, combined with 3.25, yields that lim R →∞ I ∞ R exists. Thus, for any small ε>0, there exists a large positive constant which still is denoted by R 0 , such that, for R>R 0 , lim t →∞  B 2R \B R  x  σ u  x, t  p dx ≡ I ∞ 2R − I ∞ R <ε. 3.26 Hence, by similar argument as that in 3.24, there exists a large positive number TR > 1 such that  B 2R \B R  x  σ u  x, t  p dx < ε, ∀t>T  R  . 3.27 On the other hand, we argue as in 6, 10.Letξx ∈ C 2 R N  be a positive function satisfying. i 0 ≤ ξx ≤ 1inR N ; ξx ≡ 1inB 1 , ξx ≡ 0inB c 2 ; ii ∂ξ/∂ν  0on∂B 2 \ B 1 ; iii for any α ∈ 0, 1, there exists a positive constant C α such that |Δξ|≤C α ξ α . Let R and TR be as defined in 3.26 and 3.27. Multiplying 3.1 by ξ R xξx/R and then integrating by parts in R N , we have d dt  R N uξ R dx   B 2R \B R u m Δξ R dx   R N  x  σ u p ξ R dx   R N fξ R dx. 3.28 8 Advances in Difference Equations By the definition of ξ R x,H ¨ older’s inequality, and 3.27, we have       B 2R \B R u m Δξ R dx      ≤ C α R 2  B 2R \B R u m ξ α R dx ≤ C α R 2   B 2R \B R  x  σ u mp  dx  1/p    B 2R \B R  x  −σq  /p  ξ αq  R dx  1/q  ≤ C α R 2   B 2R \B R  x  σ u p dx  m/p   B 2R \B R  x  −mσ/p−m dx  p−m/p ≤ Cε m/p R N−mσ/p−mp−m/p−2 ≤ Cε m/p , 3.29 where p   p/m,1/p   1/q   1, since  N − mσ p − m  p − m p − 2   N − 2  p −  N  σ  m p ≤ 0. 3.30 Let F R t  R N uξ R dx and G R t  R N x σ u p ξ R dx. Then, by making use of 3.29 and  R N fxdx ≥ δ for R>R 0 , 3.28 becomes F  R  t  ≥ G R  t  − Cε m/p  δ. 3.31 Thus, let ε be small enough such that Cε m/p ≤ δ/2, then F  R t ≥ G R tδ/2. Let t 0 >TR. By making use of H ¨ older’s inequality, we obtain that F R  t  ≤   R N  x  σ u p ξ R dx  1/p   R N  x  −σq/p ξ R dx  1/q ≤ G R  t  1/p   B 2R  x  −σ/p−1 dx  p−1/p ≤ CG R  t  1/p A σ  R  p−1/p , 3.32 where 1/p  1/q  1. Thus, we obtain that  t t 0 F R  s  p ds ≤ CA σ R p−1  t t 0 G R  s  ds ≤ CA σ  R  p−1  t t 0 F R  s  ds ≤ CA σ  R  p−1  F R  t  − F R  t 0  . 3.33 Advances in Difference Equations 9 Since F R t ≥ 0 for all t ≥ 0, we have F R  t  ≥ CA σ  R  −p1  t t 0 F R  s  p ds  F R  t 0  ≥ CA σ  R  −p1  t t 0 F R  s  p ds. 3.34 Let gt  t t 0 F R s p ds, then g  t   F R  t  p ≥ CA σ  R  −pp−1 g  t  p . 3.35 Let t 1 >t 0 such that gt 1  > 0. Since p>1, by solving the differential inequality 3.35 in t 1 ,t, we have  t t 1 g  s  g  s  p ds ≥ CA σ  R  −pp−1  t t 1 ds, g  t  1−p ≤ g  t 1  1−p − C  p − 1  A σ  R  −pp−1  t − t 1  , g  t  ≥  g  t 1  1−p − C  p − 1  A σ  R  −pp−1  t − t 1   −1/p−1 . 3.36 Thus, there exists T 1 with t 1 <T 1 ≤ t 1  Cp − 1 −1 A σ R pp−1 gt 1  1−p , such that lim t↑T 1 gt ∞, which implies that gt and then u blow up in finite time. It contradicts our assumption. Therefore, every positive solution of Problem 3.1 blows up in finite time. Hence, every positive solution of Problem 1.1 blows up in finite time. 4. Proof of Theorem 1.1(b) In this section, we prove that for p>mN σ/N −2, there exist some fx and u 0 x, such that Problem 1.1 admits a global positive solution. We first consider the stationary problem of Problem 1.1 as follows: Δu m   x  σ u p  f  x   0  x ∈ R N  . 4.1 Let vxC 1 x −s , where s 2  σ/p − m and the positive constant C 1 satisfies C p−m 1  ms  N − ms − 2   m  2  σ    N − 2  p −  N  σ  m   p − m  2 > 0. 4.2 10 Advances in Difference Equations Then, we have −Δv m  ms 2 C m 1  | x | 2  1  −ms/2−1 Δ  | x | 2  1  − ms  ms  2  4 C m 1  | x | 2  1  −ms/2−2    ∇  | x | 2  1     2  NmsC m 1  x  −ms−2 − ms  ms  2  C m 1 | x | 2  x  −ms−4  ms  N − ms − 2  C m 1  x  −ms−2  ms  ms  2  C m 1  x  −ms−4 . 4.3 Since C p−m 1  msN − ms − 2 and −ms − 2  σ − ps, we have −Δv m  C p 1  x  σ−ps  C 2  x  −ms−4   x  σ v p  C 2  x  −ms−4 , 4.4 where C 2  msms  2C m 1 .Thus,iffx ≤ C 2 x −ms−4 and u 0 x ≤ vx, then v is a supersolution of Problem 1.1. It is obvious that 0 is s sub-solution of Problem 1.1. Therefore, by the iterative process and the comparison theorem, Problem 1.1 admits a global positive solution. Acknowledgments This paper was introduced to the author by Professor Kiyoshi Mochizuki in Chuo University. The author would like to thank him for his proper guidance. The author would also like to thank Ryuichi Suzuki for useful discussions and friendly encouragement during the preparation of this paper. References 1 P. Baras and R. Kersner, “Local and global solvability of a class of semilinear parabolic equations,” Journal of Differential Equations, vol. 68, no. 2, pp. 238–252, 1987. 2 K. Mochizuki and K. Mukai, “Existence and nonexistence of global solutions to fast diffusions with source,” Methods and Applications of Analysis, vol. 2, no. 1, pp. 92–102, 1995. 3 R. G. Pinsky, “Existence and nonexistence of global solutions for u t Δu  axu p in R d ,” Journal of Differential Equations, vol. 133, no. 1, pp. 152–177, 1997. 4 Y W. Qi, “On the equation u t Δu α  u β ,” Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 123, no. 2, pp. 373–390, 1993. 5 R. Suzuki, “Existence and nonexistence of global solutions of quasilinear parabolic equations,” Journal of the Mathematical Society of Japan, vol. 54, no. 4, pp. 747–792, 2002. 6 C. Bandle, H. A. Levine, and Q. S. Zhang, “Critical exponents of Fujita type for inhomogeneous parabolic equations and systems,” Journal of Mathematical Analysis and Applications, vol. 251, no. 2, pp. 624–648, 2000. 7 M. Kirane and M. Qafsaoui, “Global nonexistence for the Cauchy problem of some nonlinear reaction- diffusion systems,” Journal of Mathematical Analysis and Applications, vol. 268, no. 1, pp. 217–243, 2002. 8 X. Zeng, “The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1408–1424, 2007. [...]...Advances in Difference Equations 11 9 Q S Zhang, “Blow-up results for nonlinear parabolic equations on manifolds,” Duke Mathematical Journal, vol 97, no 3, pp 515–539, 1999 10 Y.-W Qi, The critical exponents of parabolic equations and blow-up in Rn ,” Proceedings of the Royal Society of Edinburgh Section A, vol 128, no 1, pp 123–136, 1998 . Difference Equations Volume 2010, Article ID 451619, 11 pages doi:10.1155/2010/451619 Research Article Existence and Nonexistence of Global Solutions of the Quasilinear Parabolic Equations with Inhomogeneous. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 123, no. 2, pp. 373–390, 1993. 5 R. Suzuki, Existence and nonexistence of global solutions of quasilinear parabolic equations, ”. obtained. We will prove Theorem 1.1a and b in Sections 3 and 4, respectively. The proof of Theorem 1.2 is included in the proof of Theorem 1.1a. In the following, R and T are two given positive