메뉴 건너뛰기




Volumn 64, Issue , 2005, Pages 473-495

The influence of gradient perturbations on blow-up asymptotics in semilinear parabolic problems: A survey

Author keywords

Gradient perturbation; Gradient term; Nonlinear heat equation; Nonlinear parabolic equation; Quasilinear heat equation

Indexed keywords


EID: 68249093943     PISSN: 14211750     EISSN: 23740280     Source Type: Book Series    
DOI: 10.1007/3-7643-7385-7_28     Document Type: Chapter
Times cited : (14)

References (58)
  • 1
    • 84971116414 scopus 로고
    • On the blow up of solutions for a convective reaction diffusion equation
    • J. Aguirre, M. Escobedo, On the blow up of solutions for a convective reaction diffusion equation, Proc. Roy. Soc. Edinburgh., 123A, 3 (1993), 433–460.
    • (1993) Proc. Roy. Soc. Edinburgh. , vol.123A , Issue.3 , pp. 433-460
    • Aguirre, J.1    Escobedo, M.2
  • 2
    • 84968519124 scopus 로고
    • The formation of the dead core in parabolic reaction-diffusion problems
    • C. Bandle, I. Stakgold, The formation of the dead core in parabolic reaction-diffusion problems, Trans. Amer. Math. Soc., 286 (1984), 275–293.
    • (1984) Trans. Amer. Math. Soc. , vol.286 , pp. 275-293
    • Bandle, C.1    Stakgold, I.2
  • 3
    • 0001590177 scopus 로고
    • max for the solution of a semilinear heat equation
    • max for the solution of a semilinear heat equation, J. Funct. Anal., 71 (1987), 142–174.
    • (1987) J. Funct. Anal. , vol.71 , pp. 142-174
    • Baras, P.1    Cohen, L.2
  • 5
    • 1642367710 scopus 로고    scopus 로고
    • Gradient bounds for solutions of semilinear parabolic equations without Bernstein’s quadratic condition
    • J.-Ph. Bartier, Ph. Souplet, Gradient bounds for solutions of semilinear parabolic equations without Bernstein’s quadratic condition, C.R. Acad. Sc. Paris, 338 (2004), 533–538.
    • (2004) C.R. Acad. Sc. Paris , vol.338 , pp. 533-538
    • Bartier, J.-P.1    Souplet, P.2
  • 6
    • 0024719715 scopus 로고
    • Characterization of blow-up for a semilinear heat equation with a convection term
    • J. Bebernes, D. Eberly, Characterization of blow-up for a semilinear heat equation with a convection term, Quart. J. Mech. Appl. Mech., 42 (1989) 447–456.
    • (1989) Quart. J. Mech. Appl. Mech. , vol.42 , pp. 447-456
    • Bebernes, J.1    Eberly, D.2
  • 8
    • 0000877190 scopus 로고
    • Universality in blow-up for nonlinear heat equations
    • J. Bricmont, A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7 (1994), 539–575.
    • (1994) Nonlinearity , vol.7 , pp. 539-575
    • Bricmont, J.1    Kupiainen, A.2
  • 9
    • 0000581856 scopus 로고
    • Some blow up results for a nonlinear parabolic problem with a gradient term
    • M. Chipot, F.B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal., 20 (1989), 886–907.
    • (1989) SIAM J. Math. Anal. , vol.20 , pp. 886-907
    • Chipot, M.1    Weissler, F.B.2
  • 10
    • 0002499851 scopus 로고    scopus 로고
    • From critical exponents to blowup rates for parabolic problems
    • M. Chlebik, M. Fila, From critical exponents to blowup rates for parabolic problems, Rend. Mat. Appl., Ser. VII, 19 (1999), 449–470.
    • (1999) Rend. Mat. Appl., Ser , vol.7 , Issue.19 , pp. 449-470
    • Chlebik, M.1    Fila, M.2
  • 11
    • 2942549225 scopus 로고    scopus 로고
    • Blowup of positive solutions of a semilinear parabolic equation with a gradient term
    • M. Chlebik, M. Fila, P. Quittner, Blowup of positive solutions of a semilinear parabolic equation with a gradient term, Dyn. Contin. Discrete Impulsive Syst. Ser. A, 10 (2003), 525–537.
    • (2003) Dyn. Contin. Discrete Impulsive Syst. Ser. A , vol.10 , pp. 525-537
    • Chlebik, M.1    Fila, M.2    Quittner, P.3
  • 12
    • 51249168390 scopus 로고
    • Stabilization of solutions of a nonlinear parabolic equation with a gradient term
    • K. Deng, Stabilization of solutions of a nonlinear parabolic equation with a gradient term, Math. Z., 216 (1994), 147–155.
    • (1994) Math. Z. , vol.216 , pp. 147-155
    • Deng, K.1
  • 13
    • 84968503362 scopus 로고
    • Remarks on blow up for a nonlinear parabolic equation with a gradient term
    • M. Fila, Remarks on blow up for a nonlinear parabolic equation with a gradient term, Proc. Amer. Math. Soc., 111 (1991), 795–801.
    • (1991) Proc. Amer. Math. Soc. , vol.111 , pp. 795-801
    • Fila, M.1
  • 16
    • 0038563718 scopus 로고    scopus 로고
    • The blow-up rate for semilinear parabolic problems on general domains
    • M. Fila, Ph. Souplet, The blow-up rate for semilinear parabolic problems on general domains, Nonlinear Differ. Eq. and Appl. NoDEA, 8 (2001), 473–480.
    • (2001) Nonlinear Differ. Eq. and Appl. Nodea , vol.8 , pp. 473-480
    • Fila, M.1    Souplet, P.2
  • 17
    • 0010816767 scopus 로고
    • Quenching profiles for one-dimensional semilinear heat equations
    • S. Filippas, J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations, Quart. Appl. Math., 51 (1993), 713–729.
    • (1993) Quart. Appl. Math. , vol.51 , pp. 713-729
    • Filippas, S.1    Guo, J.-S.2
  • 18
    • 0011234286 scopus 로고    scopus 로고
    • Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity
    • S. Filippas, M.A. Herrero, J.J.L. Velázquez, Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity, Proc. R. Soc. Lond. A, 456 (2000), 2957–2982.
    • (2000) Proc. R. Soc. Lond. A , vol.456 , pp. 2957-2982
    • Filippas, S.1    Herrero, M.A.2    Velázquez, J.J.L.3
  • 19
    • 0023366191 scopus 로고
    • Extinction properties of semilinear heat equations with strong absorption
    • A. Friedman, M.A. Herrero, Extinction properties of semilinear heat equations with strong absorption, J. Math. Anal. Appl., 124 (1987), 530–546.
    • (1987) J. Math. Anal. Appl. , vol.124 , pp. 530-546
    • Friedman, A.1    Herrero, M.A.2
  • 20
    • 0000546335 scopus 로고
    • Blowup of positive solutions of semilinear heat equations
    • A. Friedman, J.B. McLeod, Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425–447.
    • (1985) Indiana Univ. Math. J. , vol.34 , pp. 425-447
    • Friedman, A.1    McLeod, J.B.2
  • 21
    • 0010505120 scopus 로고
    • Application of new comparison theorems to the investigation of unbounded solutions of nonlinear parabolic equations (Russian)
    • (Engl. transl.: Differ. Equations, 22 (1986), 809–815.)
    • V.A. Galaktionov, S. Posashkov, Application of new comparison theorems to the investigation of unbounded solutions of nonlinear parabolic equations (Russian), Differ. Uravn., 22 (1986), 1165–1173. (Engl. transl.: Differ. Equations, 22 (1986), 809–815.)
    • (1986) Differ. Uravn. , vol.22 , pp. 1165-1173
    • Galaktionov, V.A.1    Posashkov, S.2
  • 22
    • 21844506741 scopus 로고
    • Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations
    • V.A. Galaktionov, J.L. Vázquez, Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations, Arch. Rational Mech. Anal., 129 (1995), 225–244.
    • (1995) Arch. Rational Mech. Anal. , vol.129 , pp. 225-244
    • Galaktionov, V.A.1    Vázquez, J.L.2
  • 23
    • 0030147090 scopus 로고    scopus 로고
    • Blowup for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations
    • V.A. Galaktionov, J.L. Vázquez, Blowup for quasilinear heat equations described by means of nonlinear Hamilton-Jacobi equations, J. Differ. Equations, 127 (1996), 1–40.
    • (1996) J. Differ. Equations , vol.127 , pp. 1-40
    • Galaktionov, V.A.1    Vázquez, J.L.2
  • 24
    • 1842692780 scopus 로고    scopus 로고
    • The problem of blow-up in nonlinear parabolic equations
    • V.A. Galaktionov, J.L. Vázquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst., 8 (2002), 399–433.
    • (2002) Discrete Contin. Dyn. Syst. , vol.8 , pp. 399-433
    • Galaktionov, V.A.1    Vázquez, J.L.2
  • 25
    • 0001464964 scopus 로고
    • A bound for global solutions of semilinear heat equations
    • Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys., 103 (1986), 415–421.
    • (1986) Comm. Math. Phys. , vol.103 , pp. 415-421
    • Giga, Y.1
  • 26
    • 84990616610 scopus 로고
    • Asymptotically self-similar blow-up of semilinear heat equations
    • Y. Giga, R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297–319.
    • (1985) Comm. Pure Appl. Math. , vol.38 , pp. 297-319
    • Giga, Y.1    Kohn, R.V.2
  • 27
    • 0000332576 scopus 로고
    • Characterizing blowup using similarity variables
    • Y. Giga, R.V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1–40.
    • (1987) Indiana Univ. Math. J. , vol.36 , pp. 1-40
    • Giga, Y.1    Kohn, R.V.2
  • 28
    • 84990575181 scopus 로고
    • Nondegeneracy of blowup for semilinear heat equations
    • Y. Giga, R.V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845–884.
    • (1989) Comm. Pure Appl. Math. , vol.42 , pp. 845-884
    • Giga, Y.1    Kohn, R.V.2
  • 29
    • 3042688935 scopus 로고    scopus 로고
    • Blow up rate for semilinear heat equations with subcritical nonlinearity
    • Y. Giga, S. Matsui, S. Sasayama, Blow up rate for semilinear heat equations with subcritical nonlinearity, Indiana Univ. Math. J., 53 (2004), 483–514.
    • (2004) Indiana Univ. Math. J. , vol.53 , pp. 483-514
    • Giga, Y.1    Matsui, S.2    Sasayama, S.3
  • 30
    • 4744344778 scopus 로고    scopus 로고
    • On blow-up rate for sign-changing solutions in a convex domain
    • Y. Giga, S. Matsui, S. Sasayama, On blow-up rate for sign-changing solutions in a convex domain, Math. Methods Appl. Sci., 27 (2004), 1771–1782.
    • (2004) Math. Methods Appl. Sci. , vol.27 , pp. 1771-1782
    • Giga, Y.1    Matsui, S.2    Sasayama, S.3
  • 31
    • 85045504394 scopus 로고    scopus 로고
    • Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up
    • J.-S. Guo, Ph. Souplet, Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up, Math. Ann., to appear.
    • Math. Ann., to Appear
    • Guo, J.-S.1    Souplet, P.2
  • 32
    • 0000172301 scopus 로고
    • Explosion de solutions des équations paraboliques semilinéaires supercritiques
    • M.A. Herrero, J.J.L. Velazquez, Explosion de solutions des équations paraboliques semilinéaires supercritiques, C. R. Acad. Sc. Paris Série I, 319 (1994), 141– 145.
    • (1994) C. R. Acad. Sc. Paris Série I , vol.319 , pp. 141-145
    • Herrero, M.A.1    Velazquez, J.J.L.2
  • 33
    • 0001091789 scopus 로고    scopus 로고
    • Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition
    • B. Hu, Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition, Differ. Integral Equations, 9 (1996), 891–901.
    • (1996) Differ. Integral Equations , vol.9 , pp. 891-901
    • Hu, B.1
  • 34
    • 0000885478 scopus 로고
    • Remarks on blowup and dead cores for nonlinear parabolic equations
    • B. Kawohl, L. Peletier, Remarks on blowup and dead cores for nonlinear parabolic equations, Math. Z., 202 (1989), 207–217.
    • (1989) Math. Z. , vol.202 , pp. 207-217
    • Kawohl, B.1    Peletier, L.2
  • 35
    • 84974378755 scopus 로고
    • Global blowup of a nonlinear heat equation
    • A.A. Lacey, Global blowup of a nonlinear heat equation, Proc. Royal Soc. Edinburgh, 104 (1986), 161–167.
    • (1986) Proc. Royal Soc. Edinburgh , vol.104 , pp. 161-167
    • Lacey, A.A.1
  • 37
    • 0005885270 scopus 로고
    • A blowup result for the critical exponent in cones
    • H.A. Levine, P. Meier, A blowup result for the critical exponent in cones, Israel J. Math., 67 (1989), 129–136.
    • (1989) Israel J. Math. , vol.67 , pp. 129-136
    • Levine, H.A.1    Meier, P.2
  • 39
    • 7044224827 scopus 로고    scopus 로고
    • On nonexistence of type II blow-up for a supercritical nonlinear heat equation
    • H. Matano, F. Merle, On nonexistence of type II blow-up for a supercritical nonlinear heat equation, Commun. Pure Appl. Math., 57 (2004), 1494–1541.
    • (2004) Commun. Pure Appl. Math. , vol.57 , pp. 1494-1541
    • Matano, H.1    Merle, F.2
  • 40
    • 7044220297 scopus 로고    scopus 로고
    • Self-similar blow up patterns in supercritical semilinear heat equations
    • J. Matos, Self-similar blow up patterns in supercritical semilinear heat equations, Commun. Appl. Anal., 5 (2001), 455–483.
    • (2001) Commun. Appl. Anal. , vol.5 , pp. 455-483
    • Matos, J.1
  • 41
    • 0142259302 scopus 로고    scopus 로고
    • Universal blow-up rates for a semilinear heat equation and applications
    • J. Matos, Ph. Souplet, Universal blow-up rates for a semilinear heat equation and applications, Advances Differ. Equations, 8 (2003), 615–639.
    • (2003) Advances Differ. Equations , vol.8 , pp. 615-639
    • Matos, J.1    Souplet, P.2
  • 42
    • 0042515249 scopus 로고    scopus 로고
    • Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition
    • N. Mizoguchi, Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition, J. Differ. Equations, 193 (2003), 212–238.
    • (2003) J. Differ. Equations , vol.193 , pp. 212-238
    • Mizoguchi, N.1
  • 43
    • 0003187917 scopus 로고
    • Single point blow-up for a general semilinear heat equation
    • C. Mueller, F.B. Weissler, Single point blow-up for a general semilinear heat equation, Indiana Univ. Math. J., 34 (1985), 881–913.
    • (1985) Indiana Univ. Math. J. , vol.34 , pp. 881-913
    • Mueller, C.1    Weissler, F.B.2
  • 44
    • 84951416437 scopus 로고
    • Existence of solutions of quenching problems
    • D. Phillips, Existence of solutions of quenching problems, Appl. Anal., 24 (1987), 253–264.
    • (1987) Appl. Anal. , vol.24 , pp. 253-264
    • Phillips, D.1
  • 45
    • 85045527731 scopus 로고    scopus 로고
    • unpublished manuscript
    • M. Pierre, unpublished manuscript (2003).
    • (2003)
    • Pierre, M.1
  • 46
    • 0041737982 scopus 로고    scopus 로고
    • Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems
    • P. Quittner, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math., 29 (2003), 757–799.
    • (2003) Houston J. Math. , vol.29 , pp. 757-799
    • Quittner, P.1
  • 48
    • 0038784323 scopus 로고    scopus 로고
    • A priori estimates of global solutions of superlinear parabolic problems without variational structure
    • P. Quittner, Ph. Souplet, A priori estimates of global solutions of superlinear parabolic problems without variational structure, Discrete Contin. Dyn. Syst., 9 (2003), 1277–1292.
    • (2003) Discrete Contin. Dyn. Syst. , vol.9 , pp. 1277-1292
    • Quittner, P.1    Souplet, P.2
  • 49
    • 52549103290 scopus 로고    scopus 로고
    • Recent results and open problems on parabolic equations with gradient nonlinearities
    • Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electronic J. Differ. Equations, Vol. 2001 (2001), No. 20, pp. 1–19.
    • (2001) Electronic J. Differ. Equations , vol.2001 , Issue.20 , pp. 1-19
    • Souplet, P.1
  • 50
    • 0005794877 scopus 로고    scopus 로고
    • Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions
    • Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differ. Integral Equations, 15 (2002), 237–256.
    • (2002) Differ. Integral Equations , vol.15 , pp. 237-256
    • Souplet, P.1
  • 52
    • 0347340615 scopus 로고    scopus 로고
    • Blow-up rates for nonlinear heat equations with gradient terms and for parabolic inequalities
    • Ph. Souplet, S. Tayachi, Blow-up rates for nonlinear heat equations with gradient terms and for parabolic inequalities, Colloq. Math., 88 (2001), 135–154.
    • (2001) Colloq. Math. , vol.88 , pp. 135-154
    • Souplet, P.1    Tayachi, S.2
  • 53
    • 0000126964 scopus 로고    scopus 로고
    • Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term
    • Ph. Souplet, S. Tayachi, F.B. Weissler, Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., 48 (1996), 655–682.
    • (1996) Indiana Univ. Math. J. , vol.48 , pp. 655-682
    • Souplet, P.1    Tayachi, S.2    Weissler, F.B.3
  • 54
    • 0031195373 scopus 로고    scopus 로고
    • Self-similar sub-solutions and blow-up for nonlinear parabolic equations
    • Ph. Souplet, F.B. Weissler, Self-similar sub-solutions and blow-up for nonlinear parabolic equations, J. Math. Anal. Appl., 212 (1997), 60–74.
    • (1997) J. Math. Anal. Appl. , vol.212 , pp. 60-74
    • Souplet, P.1    Weissler, F.B.2
  • 55
    • 0012103230 scopus 로고    scopus 로고
    • Poincaré’s inequality and global solutions of a nonlinear parabolic equation
    • Ph. Souplet, F.B. Weissler, Poincaré’s inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincaré, Analyse non linéaire, 16 (1999), 337–373.
    • (1999) Ann. Inst. H. Poincaré, Analyse Non linéaire , vol.16 , pp. 337-373
    • Souplet, P.1    Weissler, F.B.2
  • 57
    • 38249006022 scopus 로고
    • The Lipschitz continuity of the interface in the heat equation with strong absorption
    • H.-M. Yin, The Lipschitz continuity of the interface in the heat equation with strong absorption, Nonlinear Anal., 20 (1993), 413–416.
    • (1993) Nonlinear Anal , vol.20 , pp. 413-416
    • Yin, H.-M.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.