Abstract and Applied Analysis

Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

Abstract

We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions: $(b(u){)}_{t}=\nabla ·(h(t)k(x)a(u)\nabla u)+f(x,u,|\nabla u{|}^{2},t)$, in $D{\times}(0,T)$, $(\partial u/\partial n)+\gamma u=0$, on $\partial D{\times}(0,T)$, $u(x,0)={u}_{0}(x)>0$, in $\overline{D}$, where $D\subset {\mathbb{R}}^{N} (N\ge 2)$ is a bounded domain with smooth boundary $\partial D$. Under some appropriate assumption on the functions $f$, $h$, $k$, $b$, and $a$ and initial value ${u}_{0}$, we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 241650, 9 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425048222

Digital Object Identifier
doi:10.1155/2014/241650

Mathematical Reviews number (MathSciNet)
MR3266300

Zentralblatt MATH identifier
07021982

Citation

Zhang, Lingling; Wang, Hui. Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 241650, 9 pages. doi:10.1155/2014/241650. https://projecteuclid.org/euclid.aaa/1425048222

References

• A. Constantin and J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,” Communications on Pure and Applied Mathematics, vol. 51, no. 5, pp. 475–504, 1998.
• J. M. Chadam, A. Peirce, and H. Yin, “The blowup property of solutions to some diffusion equations with localized nonlinear reactions,” Journal of Mathematical Analysis and Applications, vol. 169, no. 2, pp. 313–328, 1992.
• J. Wang, Z. J. Wang, and J. X. Yin, “A class of degenerate diffusion equations with mixed boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 589–603, 2004.
• N. Wolanski, “Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary,” SIAM Journal on Mathematical Analysis, vol. 24, no. 2, pp. 317–326, 1993.
• J. Ding, “Blow-up of solutions for a class of semilinear reaction diffusion equations with mixed boundary conditions,” Applied Mathematics Letters, vol. 15, no. 2, pp. 159–162, 2002.
• L. Zhang, “Blow-up of solutions for a class of nonlinear parabolic equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 25, no. 4, pp. 479–486, 2006.
• L. A. Caffarelli and A. Friedman, “Blowup of solutions of nonlinear heat equations,” Journal of Mathematical Analysis and Applications, vol. 129, no. 2, pp. 409–419, 1988.
• V. A. Galaktionov and J. L. Vázquez, “The problem of blow-up in nonlinear parabolic equations,” Discrete and Continuous Dynamical Systems, vol. 8, no. 2, pp. 399–433, 2002.
• H. Zhang and X. Guo, “Blow-up for nonlinear heat equations with absorptions,” European Journal of Pure and Applied Mathematics, vol. 1, no. 3, pp. 33–39, 2008.
• A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-Up in Problems for Quasilinear Parabolic Equations, Nauka, Moscow, Russia, 1987, (Russian), English Translation: Walter de Gruyter, Berlin, Germany, 1995.
• L. E. Payne and P. W. Schaefer, “Lower bounds for blow-up time in parabolic problems under Neumann conditions,” Applicable Analysis, vol. 85, no. 10, pp. 1301–1311, 2006.
• H. Amann, “Quasilinear parabolic systems under nonlinear boundary conditions,” Archive for Rational Mechanics and Analysis, vol. 92, no. 2, pp. 153–192, 1986.
• J. Ding and S. Li, “Blow-up solutions for a class of nonlinear parabolic equations with mixed boundary conditions,” Journal of Systems Science and Complexity, vol. 18, no. 2, pp. 265–276, 2005.
• J. Ding, “Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions,” Computers & Mathematics with Applications, vol. 65, no. 11, pp. 1808–1822, 2013.
• C. Enache, “Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition,” Applied Mathematics Letters, vol. 24, no. 3, pp. 288–292, 2011.
• H. Zhang, “Blow-up solutions and global solutions for nonlinear parabolic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4567–4574, 2008.
• R. P. Sperb, Maximum Principles and Their Applications, vol. 157 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1981. \endinput