# American Institute of Mathematical Sciences

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

 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.
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. [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. [3] H. Kielhöfer, Halbgruppen und semilineare Anfangs-randwert-probleme, Manuscripta Math.12, (1974), 121-152. [4] H.A. Levine, The role of the critical exponents in blow-up theorems, SIAM Review 32, (1990), 262-288. [5] M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim. 32, (2011), 453- 468. [6] M. Marras, S.Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete and Continuous Dynamical Systems 32, N. 11, (2012) 4001-4014. [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. [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. [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. [10] L.E. Payne, G.A. Philippin, P.W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Analysis. 69 (2008), 3495-3502. [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. [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. [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. [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. [15] P. Quittner, P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts, Basel, (2007). [16] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. [17] F.B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J. 29 (1980), 79-102 . [18] F.B. Weissler, Existence and nonexistence of global solutions for a heat equation, Israel J.Math. 38 (1981), n.1-2, 29-40.

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##### References:
 [1] J.M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford 28, (1977), 473-486. [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. [3] H. Kielhöfer, Halbgruppen und semilineare Anfangs-randwert-probleme, Manuscripta Math.12, (1974), 121-152. [4] H.A. Levine, The role of the critical exponents in blow-up theorems, SIAM Review 32, (1990), 262-288. [5] M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim. 32, (2011), 453- 468. [6] M. Marras, S.Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete and Continuous Dynamical Systems 32, N. 11, (2012) 4001-4014. [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. [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. [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. [10] L.E. Payne, G.A. Philippin, P.W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Analysis. 69 (2008), 3495-3502. [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. [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. [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. [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. [15] P. Quittner, P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Advanced Texts, Basel, (2007). [16] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. [17] F.B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J. 29 (1980), 79-102 . [18] F.B. Weissler, Existence and nonexistence of global solutions for a heat equation, Israel J.Math. 38 (1981), n.1-2, 29-40.
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