# American Institute of Mathematical Sciences

November  2014, 19(9): 3019-3029. doi: 10.3934/dcdsb.2014.19.3019

## Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source

 1 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

Received  May 2013 Revised  September 2013 Published  September 2014

Throughout this paper, we consider the equation $u_t - \Delta u = e^{|\nabla u|}$ with homogeneous Dirichlet boundary condition. One of our main goals is to show that the existence of global classical solution can derive the existence of classical stationary solution, and the global solution must converge to the stationary solution in $C(\overline{\Omega})$. On the contrary, the existence of the stationary solution also implies the global existence of the classical solution at least in the radial case. The other one is to show that finite time gradient blowup will occur for large initial data or domains with small measure.
Citation: Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019
##### References:
 [1] J. M. Arrieta, A. Rodriguez-Bernal and Ph. Souplet, Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena, Ann. Scuola. Norm. Super. Pisa Cl. Sci., 3 (2004), 1-15. [2] M. Fila and G. M. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations, 7 (1994), 811-821. [3] M. Fila, J. Taskinen and M. Winkler, Convergence to a singular steady state of a parabolic equation with gradient blow-up, Appl. Math. Lett., 20 (2007), 578-582. doi: 10.1016/j.aml.2006.07.004. [4] J.-S. Guo and B. Hu, Blowup rate estimates for the heat equation with a nonlinear gradient source term, Discrete Contin. Dyn. Syst., 20 (2008), 927-937. doi: 10.3934/dcds.2008.20.927. [5] M. Hesaaraki and A. Moameni, Blow-up of positive solutions for a family of nonlinear parabolic equations in general domain in $\mathbbR^N$, Michigan Math. J., 52 (2004), 375-389. doi: 10.1307/mmj/1091112081. [6] M. Kardar, G. Parisi and Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett., 56 (1986), 889-892. [7] J. Krug and H. Spohn, Universality classes for deterministic surface growth, Phys. Rev. A., 38 (1988), 4271-4283. doi: 10.1103/PhysRevA.38.4271. [8] Y. X. Li and Ph. Souplet, Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains, Commun. Math. Phys., 293 (2010), 499-517. doi: 10.1007/s00220-009-0936-8. [9] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 2005. [10] P. Quittner and Ph. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Verlag, Basel, 2007. [11] W. Rudin, Principles of Mathematical Analysis, $3^{rd}$ edition, McGraw-Hill, 2007. [12] Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, 15 (2002), 237-256. [13] Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electron. J. Differential Equations, 2001 (2001), 1-19. [14] Ph. Souplet and J. L. Vázquez, Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem, Discrete Contin. Dyn. Syst., 14 (2006), 221-234. [15] Ph. Souplet and Q. S. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations, J. dÁnalyse Math., 99 (2006), 335-396. doi: 10.1007/BF02789452. [16] Z. C. Zhang and B. Hu, Gradient blowup rate for a semilinear parabolic equation, Discrete Contin. Dyn. Syst., 26 (2010), 767-779. doi: 10.3934/dcds.2010.26.767. [17] Z. C. Zhang and B. Hu, Rate estimates of gradient blowup for a heat equation with exponential nonlinearity, Nonlinear Anal., 72 (2010), 4594-4601. doi: 10.1016/j.na.2010.02.036. [18] Z. C. Zhang and Y. Y. Li, Gradient blowup solutions of a semilinear parabolic equation with exponential source, Comm. Pure Appl. Anal., 12 (2013), 269-280. doi: 10.3934/cpaa.2013.12.269. [19] Z. C. Zhang and Y. Y. Li, Boundedness of global solutions for a heat equation with exponential gradient source, Abstr. Appl. Anal., 2012 (2012), 1-10. doi: doi:10.1155/2012/398049. [20] Z. C. Zhang and Z. J. Li, A note on gradient blowup rate of the inhomogeneous Hamilton-Jacobi equations, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 678-686. doi: 10.1016/S0252-9602(13)60029-6. [21] L. P. Zhu and Z. C. Zhang, Rate of approach to the steady state for a diffusion-convection equation on annular domains, Electron. J. Qual. Theory Differ. Equ., 39 (2012), 1-10.

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##### References:
 [1] J. M. Arrieta, A. Rodriguez-Bernal and Ph. Souplet, Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena, Ann. Scuola. Norm. Super. Pisa Cl. Sci., 3 (2004), 1-15. [2] M. Fila and G. M. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations, Differential Integral Equations, 7 (1994), 811-821. [3] M. Fila, J. Taskinen and M. Winkler, Convergence to a singular steady state of a parabolic equation with gradient blow-up, Appl. Math. Lett., 20 (2007), 578-582. doi: 10.1016/j.aml.2006.07.004. [4] J.-S. Guo and B. Hu, Blowup rate estimates for the heat equation with a nonlinear gradient source term, Discrete Contin. Dyn. Syst., 20 (2008), 927-937. doi: 10.3934/dcds.2008.20.927. [5] M. Hesaaraki and A. Moameni, Blow-up of positive solutions for a family of nonlinear parabolic equations in general domain in $\mathbbR^N$, Michigan Math. J., 52 (2004), 375-389. doi: 10.1307/mmj/1091112081. [6] M. Kardar, G. Parisi and Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett., 56 (1986), 889-892. [7] J. Krug and H. Spohn, Universality classes for deterministic surface growth, Phys. Rev. A., 38 (1988), 4271-4283. doi: 10.1103/PhysRevA.38.4271. [8] Y. X. Li and Ph. Souplet, Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains, Commun. Math. Phys., 293 (2010), 499-517. doi: 10.1007/s00220-009-0936-8. [9] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 2005. [10] P. Quittner and Ph. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Verlag, Basel, 2007. [11] W. Rudin, Principles of Mathematical Analysis, $3^{rd}$ edition, McGraw-Hill, 2007. [12] Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions, Differential Integral Equations, 15 (2002), 237-256. [13] Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electron. J. Differential Equations, 2001 (2001), 1-19. [14] Ph. Souplet and J. L. Vázquez, Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem, Discrete Contin. Dyn. Syst., 14 (2006), 221-234. [15] Ph. Souplet and Q. S. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations, J. dÁnalyse Math., 99 (2006), 335-396. doi: 10.1007/BF02789452. [16] Z. C. Zhang and B. Hu, Gradient blowup rate for a semilinear parabolic equation, Discrete Contin. Dyn. Syst., 26 (2010), 767-779. doi: 10.3934/dcds.2010.26.767. [17] Z. C. Zhang and B. Hu, Rate estimates of gradient blowup for a heat equation with exponential nonlinearity, Nonlinear Anal., 72 (2010), 4594-4601. doi: 10.1016/j.na.2010.02.036. [18] Z. C. Zhang and Y. Y. Li, Gradient blowup solutions of a semilinear parabolic equation with exponential source, Comm. Pure Appl. Anal., 12 (2013), 269-280. doi: 10.3934/cpaa.2013.12.269. [19] Z. C. Zhang and Y. Y. Li, Boundedness of global solutions for a heat equation with exponential gradient source, Abstr. Appl. Anal., 2012 (2012), 1-10. doi: doi:10.1155/2012/398049. [20] Z. C. Zhang and Z. J. Li, A note on gradient blowup rate of the inhomogeneous Hamilton-Jacobi equations, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 678-686. doi: 10.1016/S0252-9602(13)60029-6. [21] L. P. Zhu and Z. C. Zhang, Rate of approach to the steady state for a diffusion-convection equation on annular domains, Electron. J. Qual. Theory Differ. Equ., 39 (2012), 1-10.
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