## 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.

### MSC:

 35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian 35B44 Blow-up in context of PDEs

### Keywords:

system of semilinear parabolic equation; blow-up point
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.