##### Title: Global Existence and Blow-Up of Solutions for a Quasilinear Parabolic Equation with Absorption and Nonlinear Boundary Condition

##### Pages: 147-153

##### Cite as:

Iftikhar Ahmed, Chunlai Mu, Pan Zheng, Global Existence and Blow-Up of Solutions for a Quasilinear Parabolic Equation with Absorption and Nonlinear Boundary Condition, Int. J. Anal. Appl., 5 (2) (2014), 147-153.#### Abstract

This paper deals with the evolution r-Laplacian equation with absorption and nonlinear boundary condition. By using differential inequality techniques, global existence and blow-up criteria of nonnegative solutions are determined. Moreover, upper bound of the blow-up time for the blow-up solution is obtained.

##### Full Text: PDF

#### References

- B. Straughan, Explosive instabilities in mechanics, Springer, Berlin, 1998.
- R. Quittner and P. Souplet, Superlinear parabolic problems: blow-up, global existence and steady states, BirkhÂ¨auser Advanced Texts, Basel, 2007.
- H.A. Levine,Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: the method of unbounded Fourier cofficients, Math. Ann. 214(1975), 205-220.
- V.A. Galaktionov and J.L. Vazquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst. 8(2002), 399-433.
- Z.Q. Ling and Z.J. Wang, Global existence and blow-up for a degenerate reaction-diffusion system with nonlocal source, Appl. Math. Lett. 25(2012), 2198-2202.
- F.S. Li and J. L. Li, Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions, J. Math. Anal. Appl. 385(2012), 1005-1014.
- Y.F. Li, Y. Liu and C.H. Lin, Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions, Nonlinear Anal. RWA 11(2010), 3815-3823.
- L.E. Payne, G.A. Philippin and P.W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Anal. TMA 69(2008), 3495-3502.
- L.E. Payne, G.A. Philippin and P.W. Schaefer, Bounds for blow-up time in nonlinear parabolic problems, J. Math. Anal. Appl. 338(2008), 438-447.
- J.C. Song, Lower bounds for the blow-up time in a non-local reaction-diffusion problem, Appl. Math. Lett. 24(2011), 793-796.
- F. Liang, Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux, Nonlinear Anal. TMA 75(2012), 2189-2198.
- D.M. Liu, C.L. Mu and Q. Xin, Lower bounds estimate for the blow-up time of a nonlinear nonlocal porous medium equation, Acta Math. Sci. Ser. B Engl. Ed. 32(2012), 1206-1212.
- L.E. Payne, G.A. Philippin and S. Vernier Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I, Z. Angew. Math. Phys. 61(2010), 999-1007.
- L.E. Payne, G.A. Philippin and S. Vernier Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II, Nonlinear Anal. TMA 73(2010), 971-978.