×

Blow-up results for vector-valued nonlinear heat equations with no gradient structure. (English) Zbl 0902.35050

The author constructs a blow-up solution for the following nonlinear complex equation \[ u_{t}=\Delta u+(1+i\delta)| u| ^{p-1}u,\qquad u\in {\mathbb{C}}. \] The asymptotic profile near the singularity is obtained, as well as a generalization of the results to other vector-valued equations is proposed.

MSC:

35K40 Second-order parabolic systems
35B40 Asymptotic behavior of solutions to PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Ball, J., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. math. Oxford, Vol. 28, 473-486, (1977) · Zbl 0377.35037
[2] Berger, M.; Kohn, R., A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. pure appl. math., Vol. 41, 841-863, (1988) · Zbl 0652.65070
[3] Bricmont, J.; Kupiainen, A., Renormalization group and nonlinear pdes, () · Zbl 0842.35040
[4] Bricmont, J.; Kupiainen, A., Universality in blow-up for nonlinear heat equations, Nonlinearity, 7, 539-575, (1994) · Zbl 0857.35018
[5] Filippas, S.; Kohn, R., Refined asymptotics for the blowup of ut − δu = up, Comm. pure appl. math., Vol. 45, 821-869, (1992) · Zbl 0784.35010
[6] Filippas, S.; Merle, F., Modulation theory for the blowup of vector-valued nonlinear heat equations, J. diff. equations, Vol. 116, 119-148, (1995) · Zbl 0814.35043
[7] Galaktionov, V.A.; Kurdyumov, S.P.; Samarskii, A.A., On approximate self-similar solutions for some class of quasilinear heat equations with sources, Math. USSR-sb, Vol. 52, 155-180, (1985) · Zbl 0573.35049
[8] Galaktionov, V.A.; Vazquez, J.L., Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. math. anal., Vol. 24, 1254-1276, (1993) · Zbl 0813.35033
[9] Giga, Y.; Kohn, R., Asymptotically self-similar blowup of semilinear heat equations, Comm. pure appl. math., Vol. 38, 297-319, (1985) · Zbl 0585.35051
[10] Giga, Y.; Kohn, R., Characterizing blowup using similarity variables, Indiana univ. math. J., Vol. 36, 1-40, (1987) · Zbl 0601.35052
[11] Giga, Y.; Kohn, R., Nondegeneracy of blow-up for semilinear heat equations, Comm. pure appl. math., Vol. 42, 845-884, (1989) · Zbl 0703.35020
[12] Hamilton, R.S., The formation of singularities in the Ricci flow, (), 7-136 · Zbl 0867.53030
[13] Herrero, M.A.; Velazquez, J.J.L., Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. inst. Henri poin- caré, Vol. 10, 131-189, (1993) · Zbl 0813.35007
[14] Herrero, M.A.; Velazquez, J.J.L., Flat blow-up in one-dimensional semilinear heat equations, Differential and integral eqns., Vol. 5, 973-997, (1992) · Zbl 0767.35036
[15] Levermore, C.D.; Oliver, M., The complex Ginzburg-Landau equation as a model problem, (), 141-190 · Zbl 0845.35003
[16] Levine, H., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form put = − au + F(u), Arch. rat. mech. anal., Vol. 51, 371-386, (1973) · Zbl 0278.35052
[17] Merle, F., Solution of a nonlinear heat equation with arbitrary given blow-up points, Comm. pure appl. math., Vol. 45, 263-300, (1992) · Zbl 0785.35012
[18] {\scF. Merle} and {\scH. Zaag}, Stability of blow-up profile for equation of the type ut = Δu + |;u|;p−1u, preprint. · Zbl 0872.35049
[19] Velazquez, J.J.L., Classification of singularities for blowing up solutions in higher dimensions, Trans. amer. math. soc., Vol. 338, 441-464, (1993) · Zbl 0803.35015
[20] Weissler, F., Single-point blowup for a semilinear initial value problem, J. diff. equations, Vol. 55, 204-224, (1984) · Zbl 0555.35061
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.