LOWER BOUNDS FOR THE BLOW-UP TIME OF NONLINEAR PARABOLIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS

Transcription

1 Electroic Joural of Differetial Equatios, Vol ), No. 113, pp ISSN: URL: or ftp ejde.math.txstate.edu LOWER BOUNDS FOR THE BLOW-UP TIME OF NONLINEAR PARABOLIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS KHADIJEH BAGHAEI, MAHMOUD HESAARAKI Abstract. I this article, we fid a lower boud for the blow-up time of solutios to some oliear parabolic equatios uder Robi boudary coditios i bouded domais of R. 1. Itroductio I this article, we cosider the oliear iitial-boudary value problem bu)) t = gu) u) + fu), x, t > u + γu =, x, t >, ν ux, ) = u x), x 1.1) R, 3, is a bouded domai with smooth boudary, ν is the outward ormal vector to, γ is a positive costat ad u x) C 1 ) is the iitial value. We assume that f is a oegative CR + ) fuctio ad the oegative fuctios g ad b satisfy g C 1 R + ), gs) g m >, g s), s >, b C 2 R + ), < b s) b M, b s), s >, 1.2) g m ad b M are positive costats. The reader is referred to [1, 3, 4, 5, 6, 8] for results o bouds for blow-up time i oliear parabolic problems. Dig [2] studied problem 1.1) uder assumptios 1.2) ad derived coditios o the data which imply blow-up or the global existece of solutios. I additio, Dig obtaied a lower boud for the blow-up time whe R 3 is a bouded covex domai. Here we obtai a lower boud for the blow-up time for 1.1) i geeral bouded domais R, A lower boud for the blow-up time I this sectio we fid a lower boud for the blow-up time T i a appropriate measure. The idea of the proof of the followig theorem comes from [1]. 2 Mathematics Subject Classificatio. 35K55, 35B44. Key words ad phrases. Parabolic equatio; Robi boudary coditio; blow-up; lower boud. c 214 Texas State Uiversity - Sa Marcos. Submitted Jue 6, 213. Published April 16,

2 2 K. BAGHAEI, M. HESSARAKI EJDE-214/113 Theorem 2.1. Let be a bouded domai i R, 3, ad let the fuctios f, g, b satisfy s ) p+1 < fs) cgs) gy) dy, s >, 2.1) for some costats c > ad p 1. If ux, t) is a oegative classical solutio to problem 1.1), which becomes ubouded i the measure ux,t) ) 2k Φt) = gy) dy, k is a parameter restricted by the coditio the T is bouded from below by + Φ) k > max { p 2), 1 }, 2.2) k 1 + k 2 ξ 3 2) 3 8, 2.3) + k3 ξ 2 3 k 1, k 2 ad k 3 are positive costats which will be determied later i the proof. Proof. To simplify our computatios we defie vs) = s Hece, dt = d v 2k = 2k v 2k 1 b u) dt gu) u t = 2k v 2k 1 bu)) t gu) = 2k v 2k 1 1 [ ] gu) u) + fu) gu) = 2k2k 1) v 2k 2 v u) u 2 + 2k 2kγ v 2k 1 2k 1 fu) u ds + 2k v gu) 2k2k 1) v 2k 2 b u) gu) u 2 + 2k dy, s >. 2.4) gy) v 2k 1 g u) gu) u 2 2k 1 fu) v gu), i the above iequality we used u ad g u) from 1.2). From 2.4), we have gu) ) 2 v u 2 = 2 b. 2.5) u) By 1.2), 2.5), ad 2.1) we have 2k 2 gu) 2k2k 1) v dt b u) v 2 2k 1 fu) + 2k v gu) 22k 1)g m kb v k 2 + 2kc v 2k+p. M 2.6)

3 EJDE-214/113 LOWER BOUNDS FOR THE BLOW-UP TIME 3 From 2.2), Hölder, ad Youg iequalities, we ifer m 1 = v 2k+p m1 m 1 + m 2 v k2 3) 2 ) m2 v k2 3) 2 k2 3) 2)2k + p), m 2 = k2 3) From 2.7) ad the Cauchy-Schwartz iequality we have: ) v k2 3) 1/2 2 v 2k, 2)2k + p). k2 3) v 2k 1) 2 ) 1/2 2.7) ) 3 ) 2.8) v 2k 4 1/4. v k ) 2 2 Applyig the Sobolev iequality see [7]) to the last term i 2.8), for > 3, we obtai v k L 2 2 ) I the case, = 3, we have W 1,2 ) c s ) v k c s ) v k L 2 ) L 2 ) ) 2.9) v k L 2 2 ) c s ) v k 2 4 cs ) W 1,2 ) v k L 2 ) L 2 ) ). 2.1) Here, c s is the best costat i the Sobolev iequality. By isertig 2.9) i 2.8) for > 3 ad 2.1) i 2.8) for = 3, we have v k2 3) 2 c v 2k ) 3 4 v k L 2 ) L 2 ) ) 3 ) ) 2 3 = c v 2k 4 v k 2 4 2) + c v 2k, 2 4 cs ), for = 3, c = c s ), for > 3. Now, usig Youg s iequality we obtai v k2 3) 2 4 2) c )ɛ ) ) 2.11) 2.12) Φ 3 2) ɛ v k 2 + c Φ 2 3, 4 2)

4 4 K. BAGHAEI, M. HESSARAKI EJDE-214/113 ɛ is a positive costat to be determied later. Substitutig 2.12) ito 2.7) yields { 3 8) 2kc v 2k+p 4 2) 3 8 2kcm 2 c 4 2)ɛ Φ 3 2) ɛ v k ) } + c Φ kcm 1. By isertig the last iequality i 2.6), we have dt 22k 1)g m kb + kcm ) 2ɛ v k 2 + k 1 + k 2 Φ 3 2) 3 8 M 2 2) For k 1 = 2kcm 1, the above iequality becomes Thus, k 2 = 2kcm 2c ) 4 2) ), k 4 2)ɛ 3 = 2kcc m ɛ = 4 2)2k 1)g m k 2 cm 2 b, M dt k 1 + k 2 Φ 3 2) 3 8 k 1 + k 2 Φ 3 2) 3 8 We itegrate from to t to obtai Φt) Φ) Φ) = + k3 Φ k3 Φ 2 3 k 1 + k 2 ξ 3 2) 3 8 u x) Passig to the limit as t T, we coclude that The proof is complete. + Φ) k 1 + k 2 ξ 3 2) 3 8 Refereces + k3 ξ k3 Φ 2 3, dt. 2.13) t, gy) dy ) 2k. + k3 ξ 2 3 T. [1] A. Bao, X. Sog; Bouds for the blow-up time of the solutios to quasi-liear parabolic problems, Z. Agew. Math. Phys. 213), DOI: 1.17/s [2] J. Dig; Global ad blow-up solutios for oliear parabolic equatios with Robi boudary coditios, Comput. Math. Appl ), [3] C. Eache; Lower bouds for blow-up time i some o-liear parabolic problems uder Neuma boudary coditios, Glasg. Math. J ), [4] L. E. Paye, G. A. Philippi, P. W. Schaefer; Blow-up pheomea for some oliear parabolic problems, Noliear Aal ), [5] L. E. Paye, G. A. Philippi, P. W. Schaefer; Bouds for blow-up time i oliear parabolic problems, J. Math. Aal. Appl., ), [6] L. E. Paye, J. C. Sog, P. W. Schaefer; Lower bouds for blow-up time i a oliear parabolic problems, J. Math. Aal. Appl., ), [7] G. Taleti; Best costats i Sobolev iequality, A. Math. Pura. Appl., ),

5 EJDE-214/113 LOWER BOUNDS FOR THE BLOW-UP TIME 5 [8] H. L. Zhag; Blow-up solutios ad global solutios for oliear parabolic problems, Noliear Aal., 69 28), Khadijeh Baghaei Departmet of mathematics, Ira Uiversity of Sciece ad Techology, Tehra, Ira address: khbaghaei@iust.ac.ir Mahmoud Hesaaraki Departmet of mathematics, Sharif Uiversity of Techology, Tehra, Ira address: hesaraki@sia.sharif.edu