Abstract
This paper deals with a fourth order parabolic equation involving the Hessian, which was studied in Escudero et al. (J Math Pures Appl 103(4):924–957, 2015) recently, where the initial conditions for \(W_0^{2,2}\)-norm and \(W_0^{1,4}\)-norm blow-up were got when the initial energy \(J(u_0)\le d\), where \(d>0\) is the mountain-pass level. The purpose of this paper is to study two of the open questions proposed in the paper, that is, \(L^p\)-norm blow-up and the behavior of the solutions when \(J(u_0)>d\). For the case of \(J(u_0)<0\), we prove the solution blows up in finite time with \(L^2\)-norm. Moreover, we estimate the blow-up time and the blow-up rate. For the case of \(J(u_0)>d\), we find two sets \(\Psi _\alpha \) and \(\Phi _\alpha \), and prove that the solution blows up in finite time if the initial value belongs to \(\Psi _\alpha \), while the solution exists globally and tends to zero as time \(t\rightarrow +\infty \) when the initial value belongs to \(\Phi _\alpha \).
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References
Escudero, C.: Geometric principles of surface growth. Phys. Rev. Lett. 101(19), 196102 (2008)
Escudero, C., Peral, I.: Some fourth order nonlinear elliptic problems related to epitaxial growth. J. Differ. Equ. 254(6), 2515–2531 (2013)
Escudero, C., Hakl, R., Peral, I., Torres, P.J.: On radial stationary solutions to a model of nonequilibrium growth. Eur. J. Appl. Math. 24(3), 437–453 (2013)
Escudero, C., Hakl, R., Peral, I., Torres, P.J.: Existence and nonexistence results for a singular boundary value problem arising in the theory of epitaxial growth. Math. Methods Appl. Sci. 37(6), 793–807 (2014)
Escudero, C., Gazzola, F., Peral, I.: Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian. J. Math. Pures Appl. 103(4), 924–957 (2015)
Evans, L.: Partial differential equations, second edition. Wadsworth Brooks/cole Math. 19(1), 211–223 (2010)
Gazzola, F., Weth, T.: Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level. Differ. Integral Equ. 18(9), 961–990 (2005)
Gazzola, F., Grunau, H.C., Sweers, G.: Polyharmonic Boundary Value Problems. Springer, Berlin (2010)
Jian, Y.H., Yang, Z.D.: Bounds for the blow-up time and blow-up rate estimates for nonlinear parabolic equations with Dirichlet or Neumann boundary conditions. Br. J. Math. Comput. Sci. 12(2), 1–12 (2016)
Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889 (1986)
Luo, P.: Blow-up phenomena for a pseudo-parabolic equation. Math. Methods Appl. Sci. 38(12), 2636–2641 (2015)
Song, X.F., Lv, X.S.: Bounds for the blowup time and blowup rate estimates for a type of parabolic equations with weighted source. Appl. Math. Comput. 236(236), 78–92 (2014)
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This work is partially supported by the Basic and Advanced Research Project of CQC-STC Grant cstc2016jcyjA0018, NSFC Grant 11201380, Fundamental Research Funds for the Central Universities grant XDJK2015A16.
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Xu, G., Zhou, J. Global existence and blow-up for a fourth order parabolic equation involving the Hessian. Nonlinear Differ. Equ. Appl. 24, 41 (2017). https://doi.org/10.1007/s00030-017-0465-7
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DOI: https://doi.org/10.1007/s00030-017-0465-7