Liouville theorems and blow up behaviour in semilinear reaction diffusion systems

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This paper is concerned with positive solutions of the semilinear system: (S) {u(t) = Δu + v(p), p ≥ 1, v(t) = Δv + u(q), q ≥ 1, which blow up at x = 0 and t = T < ∞. We shall obtain here conditions on p, q and the space dimension N which yield the following bounds on the blow up rates: (1) u(x, t) ≤ C(T - t)(-p + 1/pq - 1), v(x, t) ≤ C(T - t)(-q + 1/pq - 1), for some constant C > 0. We then use (1) to derive a complete classification of blow up patterns. This last result is achieved by means of a parabolic Liouville theorem which we retain to be of some independent interest. Finally, we prove the existence of solutions of (S) exhibiting a type of asymptotics near blow up which is qualitatively different from those that hold for the scalar case.
D. Amadori, Unstable blow up patterns, to appear in Diff. and Integr. Equations. D. Andreucci, New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources, Manuscripta Math., Vol. 77, 1992, pp. 127–159. D. Andreucci, Degenerate parabolic equations with initial data measures, to appear in Trans. Amer. Mat. Soc. D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Sc. Normale Pisa, Vol. 18, 1991. J. Bebernes and A. Lacey, Finite-time blow up for a particular parabolic system, SIAM J. Math. Anal., Vol. 21, no. 6, 1990, pp. 1415–1425. J. Bebernes and A. Lacey, Finite-time blow up for semilinear reactive-diffusive systems, to appear in J. Diff. Eq. A. Bressan, On the asymptotic shape of blow up, Indiana Univ. Math. J., Vol. 39, no. 4 1990, pp. 947–960. A. Bressan, Stable blow up patterns, J. Diff. Equations, Vol. 98, 1992, pp. 57–75. J. Bricmont and A. Kupiainen, Universality in blow up for nonlinear heat equations, preprint, 1993. G. Caristi and E. Mitidieri, Blow up estimates of positive solutions of a parabolic system to appear in J. Diff. Eq. E. DiBenedetto, Degenerate parabolic equations, Springer-Verlag, New York, 1993. M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction diffusion system, J. Diff. Eq., Vol. 89, no. 1, 1991, pp. 176–202. M. Escobedo and M. A. Herrero, A semilinear parabolic system in a bounded domain Annali Mat. Pura Appl. (IV), CLXV, 1993, pp. 315–336. M. Escobedo and H. A. Levine, Critical blow up and global existence numbers for a weakly coupled system of reaction-diffusion equations, to appear in Trans. Amer. Math. Soc. S. Filippas and R. V. Kohn, Refined asymptotics for the blow up of u t −Δu=u p, Comm. Pure Appl. Math., Vol. 45, 1992, pp. 821–869. S. Filippas and F. Merle, Modulation theory for the blow up of vector valued nonlinear heat equations, to appear in J. Diff. Eq. V. A. Galaktionov and S. A. Posashkov, Application of new comparison theorems in the investigation of unbounded solutions of nonlinear parabolic equations, Diff. Urav., Vol. 22, no. 7, 1986, pp. 1165–1173. B. Gidas and J. Spruck, Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., Vol. 34, 1981, pp. 525–598. Y. Giga and R. V. Kohn, Asymptotically self-similar blow up of semilinear heat equations Comm. Pure Appl. Math., Vol. 38, 1985, pp. 297–319. Y. Giga, R.V. Kohn, Nondegeneracy of blow up for semilinear heat equations, Comm. Pure Appl. Math., Vol. 42, 1989, pp. 845–884. M. A. Herrero and J. J. L. Velázquez, Blow up behaviour of one dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré, Vol. 10, no. 2, 1993, pp. 131–189. M. A. Herrero and J. J. L. Velázquez, Flat blow up in one dimensional semilinear heat equations, Diff. and Integral Eq., Vol. 5, 1992, pp. 973–998. M. A. Herrero and J. J. L. Velázquez, Explosion de solutions d'equations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Série A, Vol. 319, 1994, pp. 141–145. O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uraltseva, Linear and quasilinear equations of parabolic type, AMS Translations of Math., Monographs, XXIII, Providence RI, 1968. F. Rothe, Global solutions of reaction-diffusion systems, in Lecture Notes in Mathematics, 1072, Springer-Verlag, New York, 1984. J. J. L. Velázquez, Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc., Vol. 338, no. 1, 1993, pp. 441–464. J. J. L. Velázquez, Higher dimensional blow up for semilinear parabolic equations, Comm. in PDE, Vol. 17, no. 9&10, 1992, pp. 1567–1596. J. J. L. Velázquez, Blow up of semilinear parabolic equations, in Recent advances in partial differential equations, eds. M. A. Herrero and E. Zuazua, Research in Applied Mathematics, Masson & J. Wiley, 1994, pp. 131–145. J. J. L. Velázquez, Curvature blow up in perturbations of minimizing cones evolving by mean curvature flow, Ann. Scuola Normale Sup. Pisa, Serie IV, Vol. XXI, 1994. pp. 595–628. F.B. Weissler, An L ∞ blow up estimate for a nolinear heat equation, Comm. Pure Appl. Math., Vol. 38, 1985, pp. 291–295.