Blow-up sets for a complex-valued semilinear heat equation. (English) Zbl 1476.35128

Summary: This paper is concerned with finite-time blow-up solutions of a one-dimensional complex-valued semilinear heat equation. We characterize the location and the number of blow-up points from the viewpoint of zeros of the solution.


35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B44 Blow-up in context of PDEs
Full Text: DOI arXiv


[1] Ackermann, N.; Bartsch, T., Superstable manifolds of semilinear parabolic problem, J. Dynam. Differential Equations, 17, 115-173, (2005) · Zbl 1129.35428
[2] Cohen, P. J.; Lees, M., Asymptotic decay of solutions of differential inequalities, Pacific J. Math., 11, 1235-1249, (1961) · Zbl 0171.35002
[3] P. Constantin, P. D. Lax, and A. Majda A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38 no. 6 (1985) 715-724. · Zbl 0615.76029
[4] Filippas, S.; Cohn, R. V., Refined asymptotics for the blowup of \({u_{t}-Δ{u}=u^{p}}\), Comm. Pure Appl. Math., 45, 821-869, (1992) · Zbl 0784.35010
[5] Fujita, H., On the blowing up of solutions of the Cauchy problem for \({u_{t}=Δ{u}+u^{1+α}}\), J. Fac. Sci. Univ. Tokyo Sect. I, 13, 109-124, (1966) · Zbl 0163.34002
[6] Guo, J. S.; Ninomiya, H.; Shimojo, M.; Yanagida, E., Convergence and blow-up of solutions for a complex-valued heat equation with a quadratic nonlinearity, Trans. Amer. Math. Soc., 365, 2447-2467, (2013) · Zbl 1277.35070
[7] Harada, J., Blowup profile for a complex valued semilinear heat equation, J. Funct. Anal., 270, 4213-4255, (2016) · Zbl 1338.35070
[8] Herrero, M. A.; Veláuez, J. J. L., Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. Partial Differential Equations, 17, 205-219, (1992) · Zbl 0772.35027
[9] Matano, H., Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29, 401-441, (1982) · Zbl 0496.35011
[10] Naito, Y.; Suzuki, T., Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity, J. Differential Equations, 232, 176-211, (2007) · Zbl 1110.35029
[11] Ogawa, H., Lower bounds for solutions of differential inequalities in Hilbert space, Proc. Amer. Math. Soc., 16, 1241-1243, (1965) · Zbl 0143.16702
[12] R. Palais Blowup for nonlinear equations using a comparison principle in Fourier space, Comm. Pure Appl. Math. 41 no. 2 (1988) 165-196. · Zbl 0674.35045
[13] Sakajo, T., Blow-up solutions of the Constantin-Lax-Majda equation with a generalized viscosity term, J. Math. Sci. Univ. Tokyo, 10, 187-207, (2003) · Zbl 1030.35006
[14] T. Sakajo, On global solutions for the Constantin-Lax-Majda equation with a generalized viscosityterm, Nonlinearity 16 no. 4 (2003) 1319-228. · Zbl 1140.76332
[15] Schochet, S., Explicit solutions of the viscous model vorticity equation, Comm. Pure Appl. Math., 39, 531-537, (1986) · Zbl 0623.76012
[16] Yang, Y., Behavior of solutions of model equations for incompressible fluid flow, J. Differential Equations, 125, 33-153, (1996)
[17] Nouaili, N.; Zaag, H., Profile for a simultaneously blowing up solution for a complex valued semilinear heat equation, Comm. Partial Differential Equations, 40, 1197-1217, (2015) · Zbl 1335.35126
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