American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 535-544. doi: 10.3934/proc.2013.2013.535

On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients

Monica Marras 1, and  Stella Vernier Piro 1,

1. 

Dipartimento di Matematica e Informatica, Università di Cagliari, 09123

Received  September 2012 Revised  December 2012 Published  November 2013

This paper deals with the blow-up of the solutions to a class of nonlinear parabolic equations with Dirichlet boundary condition and time dependent coefficients. Under some conditions on the data and geometry of the spatial domain, explicit upper and lower bounds for the blow-up time are derived. Moreover, the influence of the data on the behaviour of the solution is investigated to obtain global existence.
Keywords: blow-up solutions., Parabolic problems.
Mathematics Subject Classification: Primary: 35K55, 35K60; Secondary: 35K4.
Citation: Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535
References:
[1]

J.M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford 28, (1977), 473-486.  Google Scholar

[2]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t= \Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo 13, (1966), 109-124.  Google Scholar

[3]

H. Kielhöfer, Halbgruppen und semilineare Anfangs-randwert-probleme, Manuscripta Math.12, (1974), 121-152.  Google Scholar

[4]

H.A. Levine, The role of the critical exponents in blow-up theorems, SIAM Review 32, (1990), 262-288.  Google Scholar

[5]

M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim. 32, (2011), 453- 468.  Google Scholar

[6]

M. Marras, S.Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete and Continuous Dynamical Systems 32, N. 11, (2012) 4001-4014.  Google Scholar

[7]

M. Marras, S.Vernier Piro, Bounds for blow-up time in nonlinear parabolic system, Discrete and Continuous Dynamical Systems, Suppl. 2011 (2011) 1025-1031.  Google Scholar

[8]

L.E. Payne, G.A. Philippin, Blow up phenomena in parabolic problems with time dependent coefficients under Dirichlet boundary conditions, Proc. Amer. Math. Soc. 141, N. 7 (2013) 2309-2318.  Google Scholar

[9]

L.E. Payne, G.A. Philippin, P.W. Schaefer, Bounds for blow-up time in nonlinear parabolic problems, J.Math. Anal. Appl. 338 (2008), 438-447.  Google Scholar

[10]

L.E. Payne, G.A. Philippin, P.W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Analysis. 69 (2008), 3495-3502.  Google Scholar

[11]

L.E. Payne, G.A. Philippin, S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition I, Z. Angew. Math. Phys., 61 (2010), 971-978.  Google Scholar

[12]

L.E. Payne, G.A. Philippin, S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition II, Nonlinear Analysis, 73 (2010), 971-978.  Google Scholar

[13]

L.E. Payne, P.W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet boundary conditions, J. Math. Anal. Appl., 328 (2007), 1196-1205.  Google Scholar

[14]

L.E. Payne, P.W. Schaefer, Blow-up phenomena for some nonlinear parabolic systems, Int. J. of Pure and Applied Math., 42 (2008), 193-202.  Google Scholar

[15]

P. Quittner, P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts, Basel, (2007).  Google Scholar

[16]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  Google Scholar

[17]

F.B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J. 29 (1980), 79-102 .  Google Scholar

[18]

F.B. Weissler, Existence and nonexistence of global solutions for a heat equation, Israel J.Math. 38 (1981), n.1-2, 29-40.  Google Scholar

show all references

References:
[1]

J.M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford 28, (1977), 473-486.  Google Scholar

[2]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t= \Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo 13, (1966), 109-124.  Google Scholar

[3]

H. Kielhöfer, Halbgruppen und semilineare Anfangs-randwert-probleme, Manuscripta Math.12, (1974), 121-152.  Google Scholar

[4]

H.A. Levine, The role of the critical exponents in blow-up theorems, SIAM Review 32, (1990), 262-288.  Google Scholar

[5]

M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim. 32, (2011), 453- 468.  Google Scholar

[6]

M. Marras, S.Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete and Continuous Dynamical Systems 32, N. 11, (2012) 4001-4014.  Google Scholar

[7]

M. Marras, S.Vernier Piro, Bounds for blow-up time in nonlinear parabolic system, Discrete and Continuous Dynamical Systems, Suppl. 2011 (2011) 1025-1031.  Google Scholar

[8]

L.E. Payne, G.A. Philippin, Blow up phenomena in parabolic problems with time dependent coefficients under Dirichlet boundary conditions, Proc. Amer. Math. Soc. 141, N. 7 (2013) 2309-2318.  Google Scholar

[9]

L.E. Payne, G.A. Philippin, P.W. Schaefer, Bounds for blow-up time in nonlinear parabolic problems, J.Math. Anal. Appl. 338 (2008), 438-447.  Google Scholar

[10]

L.E. Payne, G.A. Philippin, P.W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Analysis. 69 (2008), 3495-3502.  Google Scholar

[11]

L.E. Payne, G.A. Philippin, S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition I, Z. Angew. Math. Phys., 61 (2010), 971-978.  Google Scholar

[12]

L.E. Payne, G.A. Philippin, S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition II, Nonlinear Analysis, 73 (2010), 971-978.  Google Scholar

[13]

L.E. Payne, P.W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet boundary conditions, J. Math. Anal. Appl., 328 (2007), 1196-1205.  Google Scholar

[14]

L.E. Payne, P.W. Schaefer, Blow-up phenomena for some nonlinear parabolic systems, Int. J. of Pure and Applied Math., 42 (2008), 193-202.  Google Scholar

[15]

P. Quittner, P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts, Basel, (2007).  Google Scholar

[16]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  Google Scholar

[17]

F.B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J. 29 (1980), 79-102 .  Google Scholar

[18]

F.B. Weissler, Existence and nonexistence of global solutions for a heat equation, Israel J.Math. 38 (1981), n.1-2, 29-40.  Google Scholar

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