Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions
- 110 Downloads
Abstract
Keywords
Blow-up p-Laplacian equation Nonlinear boundary conditionMSC
35K65 35B401 Introduction
We proceed as follows. In Sect. 2, we set up some conditions to ensure that the solution blows up in a finite time. An upper estimate of the blow-up solution and an upper bound of the blow-up time are also given. Section 3 is devoted to finding some conditions to guarantee that the solution exists globally. At the same time, we also obtain an upper estimate of the global solution. In Sect. 4, as applications of the abstract results, two examples are presented.
2 Blow-up solution
Theorem 2.1
- (i)$$ 0< \beta\leq\alpha; $$(2.5)
- (ii)$$ \int^{+\infty}_{M_{0}}\frac{{\mathrm{e}}^{\tau}}{g(\tau)}\, {\mathrm{d}}\tau < + \infty, \qquad M_{0}=\max_{\overline{D}}u_{0}(x); $$(2.6)
- (iii)$$ \begin{aligned} &1-2\frac{g'(s)}{g(s)}+ \frac{g''(s)}{g(s)}\geq0, \qquad (p-2) \biggl(\frac{g'(s)}{g(s)}-1 \biggr)- \frac{h''(s)}{h'(s)}\geq 0, \\ &\frac{f'(s)}{f(s)}-(p-1) \biggl(\frac{g'(s)}{g(s)}-1 \biggr)\geq0, \quad s\in \overline{\mathbb{R}_{+}}. \end{aligned} $$(2.7)
Proof
- (a)
for \(t=0\),
- (b)
at a point where \(|\nabla u|=0\),
- (c)
on the boundary \(\partial D\times(0,t^{*})\).
3 Global solution
Theorem 3.1
- (i)$$ \eta\geq\xi>0; $$(3.5)
- (ii)$$ \int^{+\infty}_{m_{0}}\frac{{\mathrm{e}}^{-\tau}}{g(\tau)}\, {\mathrm{d}}\tau =+ \infty; $$(3.6)
- (iii)$$ \begin{aligned} &1+2\frac{g'(s)}{g(s)}+ \frac{g''(s)}{g(s)}\leq0, \qquad (p-2) \biggl(\frac{g'(s)}{g(s)}+1 \biggr)- \frac{h''(s)}{h'(s)}\leq0, \\ &\frac{f'(s)}{f(s)}-(p-1) \biggl(\frac{g'(s)}{g(s)}+1 \biggr)\leq0, \quad s\in \overline{\mathbb{R}}. \end{aligned} $$(3.7)
Proof
- (a)
for \(t=0\),
- (b)
at a point where \(|\nabla u|=0\),
- (c)
on the boundary \(\partial D\times(0,t^{*})\).
4 Applications
In this section, we give two examples to illustrate the results of Theorems 2.1 and 3.1.
Example 4.1
Example 4.2
5 Conclusion
In this paper, we research the blow-up and global solutions of p-Laplacian parabolic problem (1.1). We find that it is difficult to study the existence of blow-up and global solutions of problem (1.1) by using the differential inequality technique in [1]. The main reason for this is that the boundary conditions in problems (1.1) and (1.2) are different. As in [16] and [22], we combine the parabolic maximum principle with differential inequality to study problem (1.1). The difficulty of using this method is the need to construct some appropriate auxiliary functions. Since the principal parts of the two equations are different in problems (1.1) and (1.3), the auxiliary functions in papers [16] and [20] are not suitable for problem (1.1). Therefore, the key to our study is to construct new auxiliary functions P, Φ, Q, and Ψ defined in (2.3), (2.4), (3.3), and (3.4), respectively. Using these auxiliary functions, the parabolic maximum principle, and the differential inequality technique, we complete the study of (1.1). We set up the conditions on functions f, g, h, and \(u_{0}\) to ensure that the solution of (1.1) either blows up or exists globally. In addition, an upper estimate of the global solution and the blow-up rate are obtained. We also give an upper bound for the blow-up time.
Notes
Author’s contributions
All results belong to JD. The author read and approved the final manuscript.
Competing interests
The author declares that there is no conflict of interests regarding the publication of this paper.
References
- 1.Ding, J.T., Shen, X.H.: Blow-up in p-Laplacian heat equations with nonlinear boundary conditions. Z. Angew. Math. Phys. 67, 1–18 (2016) CrossRefMATHMathSciNetGoogle Scholar
- 2.Kbiri Alaoui, M., Messaoudi, S.A., Khenous, H.B.: A blow-up result for nonlinear generalized heat equation. Comput. Math. Appl. 68, 1723–1732 (2014) CrossRefMATHMathSciNetGoogle Scholar
- 3.Li, F.S., Li, J.L.: Global existence and blow-up phenomena for p-Laplacian heat equation with inhomogeneous Neumann boundary conditions. Bound. Value Probl. 2014, 219 (2014) CrossRefMATHMathSciNetGoogle Scholar
- 4.Niculescu, C.P., Roventa, L.: Generalized convexity and the existence of finite time blow-up solutions for an evolutionary problem. Nonlinear Anal. TMA 75, 270–277 (2012) CrossRefMATHMathSciNetGoogle Scholar
- 5.Zhang, Z.C., Li, Z.J.: A universal bound for radial solutions of the quasilinear parabolic equation with p-Laplace operator. J. Math. Anal. Appl. 385, 125–134 (2012) CrossRefMATHMathSciNetGoogle Scholar
- 6.Payne, L.E., Philippin, G.A., Vernier Piro, S.: Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II. Nonlinear Anal. TMA 73, 971–978 (2010) CrossRefMATHMathSciNetGoogle Scholar
- 7.Liang, Z.L., Zhao, J.N.: Localization for the evolution p-Laplacian equation with strongly nonlinear source term. J. Differ. Equ. 246, 391–407 (2009) CrossRefMATHMathSciNetGoogle Scholar
- 8.Tersenov, A.S., Tersenov, A.S.: The problem of Dirichlet for evolution one-dimensional p-Laplacian with nonlinear source. J. Math. Anal. Appl. 340, 1109–1119 (2008) CrossRefMATHMathSciNetGoogle Scholar
- 9.Zeng, X.Z.: Blow-up results and global existence of positive solutions for the inhomogeneous evolution p-Laplacian equations. Nonlinear Anal. TMA 66, 1290–1301 (2007) CrossRefMATHMathSciNetGoogle Scholar
- 10.Li, F.C., Xie, C.H.: Global and blow-up solutions to a p-Laplacian equation with nonlocal source. Comput. Math. Appl. 46, 1525–1533 (2003) CrossRefMATHMathSciNetGoogle Scholar
- 11.Chen, C.S., Wang, R.Y.: \(L^{\infty}\) estimates of solution for the evolution m-Laplacian equation with initial value in \(L_{q} (\Omega)\). Nonlinear Anal. TMA 48, 607–616 (2002) CrossRefMATHGoogle Scholar
- 12.Friedman, A.: Partial Differential Equation of Parabolic Type. Prentice Hall, Englewood Cliffs (1964) MATHGoogle Scholar
- 13.Ma, L.W., Fang, Z.B.: Blow-up analysis for a reaction–diffusion equation with weighted nonlocal inner absorptions under nonlinear boundary flux. Nonlinear Anal., Real World Appl. 32, 338–354 (2016) CrossRefMATHMathSciNetGoogle Scholar
- 14.Ding, J.T.: Blow-up phenomena for nonlinear reaction–diffusion equations under nonlinear boundary conditions. J. Funct. Spaces 2016, Article ID 8107657 (2016) MATHMathSciNetGoogle Scholar
- 15.Harada, J.: Blow-up behavior of solutions to the heat equation with nonlinear boundary conditions. Adv. Differ. Equ. 20, 23–76 (2015) MATHMathSciNetGoogle Scholar
- 16.Zhang, L.L., Zhang, N., Li, L.X.: Blow-up solutions and global existence for a kind of quasilinear reaction–diffusion equations. Z. Anal. Anwend. 33, 247–258 (2014) CrossRefMATHMathSciNetGoogle Scholar
- 17.Baghaei, K., Hesaaraki, M.: Lower bounds for the blow-up time in the higher-dimensional nonlinear divergence form parabolic equations. C. R. Math. Acad. Sci. Paris 351, 731–735 (2013) CrossRefMATHMathSciNetGoogle Scholar
- 18.Harada, J., Mihara, K.: Blow-up rate for radially symmetric solutions of some parabolic equations with nonlinear boundary conditions. J. Differ. Equ. 253, 1647–1663 (2012) CrossRefMATHMathSciNetGoogle Scholar
- 19.Xiang, Z.Y., Wang, Y., Yang, H.Z.: Global existence and nonexistence for degenerate parabolic equations with nonlinear boundary flux. Comput. Math. Appl. 62, 3056–3065 (2011) CrossRefMATHMathSciNetGoogle Scholar
- 20.Ding, J.T., Guo, B.Z.: Global existence and blow-up solutions for quasilinear reaction-diffusion equations with a gradient term. Appl. Math. Lett. 24, 936–942 (2011) CrossRefMATHMathSciNetGoogle Scholar
- 21.Payne, L.E., Philippin, G.A., Vernier Piro, S.: Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I. Z. Angew. Math. Phys. 61, 999–1007 (2010) CrossRefMATHMathSciNetGoogle Scholar
- 22.Zhang, H.L.: Blow-up solutions and global solutions for nonlinear parabolic problems. Nonlinear Anal. TMA 69, 4567–4574 (2008) CrossRefMATHMathSciNetGoogle Scholar
- 23.Quittner, P., Rodríguez-Bernal, A.: Complete and energy blow-up in parabolic problems with nonlinear boundary conditions. Nonlinear Anal. TMA 62, 863–875 (2005) CrossRefMATHMathSciNetGoogle Scholar
- 24.Chen, W.Y.: The blow-up estimate for heat equations with non-linear boundary conditions. Appl. Math. Comput. 156, 355–366 (2004) MATHMathSciNetGoogle Scholar
- 25.Fila, M., Guo, J.S.: Complete blow-up and incomplete quenching for the heat equation with a nonlinear boundary condition. Nonlinear Anal. TMA 48, 995–1002 (2002) CrossRefMATHMathSciNetGoogle Scholar
- 26.Rodriguez-Bernal, A., Tajdine, A.: Dynamics of reaction diffusion equations under nonlinear boundary conditions. C. R. Acad. Sci. Paris, Sér. I Math. 331, 531–536 (2000) MATHMathSciNetGoogle Scholar
- 27.Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs (1967) MATHGoogle Scholar
Copyright information
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.